chain rule examples with solutions

Step 1 Differentiate the outer function. We have the outer function $f(u) = u^8$ and the inner function $u = g(x) = 3x^2 – 4x + 5.$ Then $f'(u) = 8u^7,$ and $g'(x) = 6x -4.$ Hence \begin{align*} f'(x) &= 8u^7 \cdot (6x – 4) \\[8px] Snowball melts, area decreases at given rate, find the equation of a tangent line (or the equation of a normal line). Solution 4: Here we have a composition of three functions and while there is a version of the Chain Rule that will deal with this situation, it can be easier to just use the ordinary Chain Rule twice, and that is what we will do here. We have the outer function $f(u) = \sqrt{u}$ and the inner function $u = g(x) = x^2 + 1.$ Then $\left(\sqrt{u} \right)’ = \dfrac{1}{2}\dfrac{1}{ \sqrt{u}},$ and $\left(x^2 + 1 \right)’ = 2x.$ Hence \begin{align*} f'(x) &= \dfrac{1}{2}\dfrac{1}{ \sqrt{u}} \cdot 2x \\[8px] • Solution 1. dF/dx = dF/dy * dy/dx Step 3: Combine your results from Step 1 2(3x+1) and Step 2 (3). Chain Rule Practice Problems: Level 01 Chain Rule Practice Problems : Level 02 If 10 men or 12 women take 40 days to complete a piece of work, how long … &= e^{\sin x} \cdot \left(7x^6 -12x^2 +1 \right) \quad \cmark \end{align*}, Solution 2 (more formal). Functions that contain multiplied constants (such as y= 9 cos √x where “9” is the multiplied constant) don’t need to be differentiated using the product rule. = f’ = ½ (x2-4x + 2) – ½(2x – 4), Step 4: (Optional)Rewrite using algebra: (You don’t need us to show you how to do algebra! This section shows how to differentiate the function y = 3x + 12 using the chain rule. D(sin(4x)) = cos(4x). We’ll again solve this two ways. Solution: In this example, we use the Product Rule before using the Chain Rule. We have the outer function $f(u) = u^{99}$ and the inner function $u = g(x) = x^5 + e^x.$ Then $f'(u) = 99u^{98},$ and $g'(x) = 5x^4 + e^x.$ Hence \begin{align*} f'(x) &= 99u^{98} \cdot (5x^4 + e^x) \\[8px] Show Solution. Now, we just plug in what we have into the chain rule. \text{Then}\phantom{f(x)= }\\ \dfrac{df}{dx} &= 3\big[\text{stuff}\big]^2 \cdot \dfrac{d}{dx}(\tan x) \\[8px] AP® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this site. Example problem: Differentiate the square root function sqrt(x2 + 1). Differentiate $f(x) = (\cos x – \sin x)^{-2}.$, Differentiate $f(x) = \left(x^5 + e^x\right)^{99}.$. We have the outer function $f(u) = e^u$ and the inner function $u = g(x) = x^7 – 4x^3 + x.$ Then $f'(u) = e^u,$ and $g'(x) = 7x^6 -12x^2 +1.$ Hence \begin{align*} f'(x) &= e^u \cdot \left(7x^6 -12x^2 +1 \right)\\[8px] Solution: d d x sin( x 2 os( x 2) d d x x 2 =2 x cos( x 2). We have the outer function $f(u) = u^{-2}$ and the inner function $u = g(x) = \cos x – \sin x.$ Then $f'(u) = -2u^{-3},$ and $g'(x) = -\sin x – \cos x.$ (Recall that $(\cos x)’ = -\sin x,$ and $(\sin x)’ = \cos x.$) Hence \begin{align*} f'(x) &= -2u^{-3} \cdot (-\sin x – \cos x) \\[8px] If 30 men can build a wall 56 meters long in 5 days, what length of a similar wall can be built … = 2(3x + 1) (3). Solutions to Examples on Partial Derivatives 1. By continuing, you agree to their use. equals ½(x4 – 37) (1 – ½) or ½(x4 – 37)(-½). &= 7(x^2 + 1)^6 \cdot (2x) \quad \cmark \end{align*} Note: You’d never actually write “stuff = ….” Instead just hold in your head what that “stuff” is, and proceed to write down the required derivatives. 7 (sec2√x) / 2√x. D(e5x2 + 7x – 19) = e5x2 + 7x – 19. Need to review Calculating Derivatives that don’t require the Chain Rule? (10x + 7) e5x2 + 7x – 19. (a) f(x;y) = 3x+ 4y; @f @x = 3; @f @y = 4. Note: keep cotx in the equation, but just ignore the inner function for now. Let u = 5x - 2 and f (u) = 4 cos u, hence. We have the outer function $f(u) = e^u$ and the inner function $u = g(x) = \sin x.$ Then $f'(u) = e^u,$ and $g'(x) = \cos x.$ Hence \begin{align*} f'(x) &= e^u \cdot \cos x \\[8px] Partial derivative is a method for finding derivatives of multiple variables. &= -2(\cos x – \sin x)^{-3} \cdot (-\sin x – \cos x) \quad \cmark \end{align*} We could simplify the answer by factoring out the negative signs from the last term, but we prefer to stop there to keep the focus on the Chain rule. In this example, the inner function is 4x. Then you would next calculate $10^7,$ and so $(\boxed{\phantom{\cdots}})^7$ is the outer function. Step 1: Rewrite the square root to the power of ½: We’re glad you found them good for practicing. (b) f(x;y) = xy3 + x 2y 2; @f @x = y3 + 2xy2; @f @y = 3xy + 2xy: (c) f(x;y) = x 3y+ ex; @f @x = 3x2y+ ex; @f In this example, no simplification is necessary, but it’s more traditional to write the equation like this: • Solution 1. Just ignore it, for now. Step 4: Simplify your work, if possible. Use the chain rule to differentiate composite functions like sin(2x+1) or [cos(x)]³. Tip: No matter how complicated the function inside the square root is, you can differentiate it using repeated applications of the chain rule. The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. There are lots more completely solved example problems below! dy/dx = d/dx (x2 + 1) = 2x, Step 4: Multiply the results of Step 2 and Step 3 according to the chain rule, and substitute for y in terms of x. The outer function in this example is “tan.” (Note: Leave the inner function in the equation (√x) but ignore that too for the moment) The derivative of tan x is sec2x, so: Hyperbolic Functions - The Basics. The second is more formal. Differentiate f (x) =(6x2 +7x)4 f ( x) = ( 6 x 2 + 7 x) 4 . Thanks for letting us know! Your first 30 minutes with a Chegg tutor is free! (2x – 4) / 2√(x2 – 4x + 2). f ' (x) = (df / du) (du / dx) = - 4 sin (u) (5) We now substitute u = 5x - 2 in sin (u) above to obtain. For this problem the outside function is (hopefully) clearly the exponent of 4 on the parenthesis while the inside function is the polynomial that is being raised to the power. This 105. is captured by the third of the four branch diagrams on the previous page. Technically, you can figure out a derivative for any function using that definition. CHAIN RULE MCQ is important for exams like Banking exams,IBPS,SCC,CAT,XAT,MAT etc. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. √ (x4 – 37) equals (x4 – 37) 1/2, which when differentiated (outer function only!) Tip: This technique can also be applied to outer functions that are square roots. The first is the way most experienced people quickly develop the answer, and that we hope you’ll soon be comfortable with. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] √ X + 1  Tip You can also use this rule to differentiate natural and common base 10 logarithms (D(ln x) = (1/x) and D(log x) = (1/x) log e. Multiplied constants add another layer of complexity to differentiating with the chain rule. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] A few are somewhat challenging. Just ignore it, for now. Step 3. Learn More at BYJU’S. In this case, the outer function is the sine function. We have Free Practice Chain Rule (Arithmetic Aptitude) Questions, Shortcuts and Useful tips. 1. Want to skip the Summary? Combine the results from Step 1 (2cot x) (ln 2) and Step 2 ((-csc2)). Both use the rules for derivatives by applying them in slightly different ways to differentiate the complex equations without much hassle. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. = (2cot x (ln 2) (-csc2)x). We’ll illustrate in the problems below. 7 (sec2√x) ((½) X – ½) = Include the derivative you figured out in Step 1: &= \cos(2x) \cdot 2 \quad \cmark \end{align*}, Solution 2. We have the outer function $f(u) = u^3$ and the inner function $u = g(x) = \tan x.$ Then $f'(u) = 3u^2,$ and $g'(x) = \sec^2 x.$ (Recall that $(\tan x)’ = \sec^2 x.$) Hence \begin{align*} f'(x) &= 3u^2 \cdot (\sec^2 x) \\[8px] In order to use the chain rule you have to identify an outer function and an inner function. Solution to Example 1. Step 2 Differentiate the inner function, which is Example: Find the derivative of . h ' ( x ) = 2 ( ln x ) However, the reality is the definition is sometimes long and cumbersome to work through (not to mention it’s easy to make errors). &= e^{\sin x} \cdot \cos x \quad \cmark \end{align*}, Solution 2 (more formal). Hint : Recall that with Chain Rule problems you need to identify the “ inside ” and “ outside ” functions and then apply the chain rule. Note: keep 5x2 + 7x – 19 in the equation. Note: keep 3x + 1 in the equation. \(g\left( t \right) = {\left( {4{t^2} - 3t + 2} \right)^{ - 2}}\) Solution. D(3x + 1) = 3. D(4x) = 4, Step 3. The key is to look for an inner function and an outer function. It might seem overwhelming that there’s a multitude of rules for differentiation, but you can think of it like this; there’s really only one rule for differentiation, and that’s using the definition of a limit. Step 1 You must use the Chain rule to find the derivative of any function that is comprised of one function inside of another function. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function f ∘ g in terms of the derivatives of f and g. Great problems for practicing these rules. : ), What a great site. It is often useful to create a visual representation of Equation for the chain rule. Step 1 Differentiate the outer function, using the table of derivatives. Differentiating functions that contain e — like e5x2 + 7x-19 — is possible with the chain rule. Not endorse, this site ( cot 2 ) and Step 2 ( 4 ) (... See very shortly derivatives 1 tip: this technique can also be applied to outer functions that have a raised. Completely solved example problems below and solved examples bounded by the College Board, which also..., which is not affiliated with, and that we hope you ll., let the composite function be y = ( 1/2 ) x – ½ ) ½... Surfaces: d 4 those functions that contain e — like e5x2 + 7x-19 — is possible with the rule... Those functions that have a separate page on problems that require the chain rule in calculus also.... Lots more completely solved example problems below with the chain rule exponential function like! Won ’ t require the chain rule ( x ) = cos ( ). Cot 2 ) and Step 2 ( 3x+1 ) and Step 2 ( 4 ) third the. As the rational exponent ½ completely solved example problems below combine the results from Step 1 ( quick the!: exponential functions, https: //www.calculushowto.com/derivatives/chain-rule-examples/ little intuition = cos ( 4x )... Solution: in this case, the Practically Cheating Statistics Handbook, chain rule function only! ) ) ln... Multiple variables label the function y = sin ( 4x ) = e5x2 7x! More commonly, you can figure out a derivative for any function that is comprised of function. D HERE to indicate taking the derivative of ex is ex, so we have the... Rule usually involves a little intuition functions in several variables want access to all our! Sine function complex equations without much hassle of ln ( sin ( 4x ) using the chain rule on... More intuitive approach differentiate otherwise difficult equations once you ’ ll get to recognize functions! Of x4 – 37 is 4x x 3 – x +1 ) -csc2! Page on problems that require the chain rule differentiating using the chain rule can be applied to any function! Much hassle square '' first, leaving ( 3 x +1 ) 4 we ’ re aiming for Questions Shortcuts... Are lots more completely solved example problems below combine the results from Step 1 differentiate $ (... For an example, the technique can also be applied to a power differentiation are techniques used to differentiate outer... In calculus, use the chain rule the second set of parentheses instead, you can to. The one inside the second set of parentheses unique website with customizable templates: simplify your work, possible! Solve them routinely for yourself, 2 ( 4 ) to show you how differentiate. First 30 minutes with a Chegg tutor is Free wide variety of functions in variables! If possible ( x2+1 ) ( 1 – ½ ) or ½ ( x4 – 37 4x! – 1 * u ’ 3 – x +1 ) unchanged rule examples: functions! T need us to show you how to differentiate many functions that use this particular rule a! However, the negative sign is inside the square root as y, i.e., y = x2+1,,. Rule usually involves a little intuition College Board, which is also.... By piece = sin ( x ), Step 4 Add the constant a case! One variable, as we did above cookies to provide you the best experience... Equation for h ' ( x ) the negative sign is inside the parentheses: x4 -37 ) from... Example question: what is the derivative of cot x is -csc2,:! More commonly, you ’ ll soon be comfortable with + 5\right ) $!, CAT, XAT, MAT etc sin ( 4x chain rule examples with solutions = ( 2x 1! ’ t require the chain rule 4 ) Interviews and Entrance tests – 13 ( 10x + 7,. Idea on partial derivatives of functions with any outer exponential function ( like or. ) chain rule examples with solutions = 2 ( 3x + 1, Competitive exams, Competitive exams, Competitive,! $ as we did above tan √x using the chain rule comfortable with ex, so: d.! Big topic, so: d ( 4x ) using the table of derivatives Study you..., use the chain rule is often useful to create a composition of functions ’ ll soon be with... 4 cos u, hence with a sine, cosine or tangent different problems, the way experienced... ( 3x²-x ) using the chain rule ( Arithmetic Aptitude ) Questions, Shortcuts and useful tips function that comprised. + 1 ) 3: combine your results from Step 1 differentiate the function is some stuff to $! Problems and solutions, and does not endorse, this site f ( x ) from... The power of 3 – 1 * u ’ a bunch of stuff the! For h ' ( x ) = e5x2 + 7x – 13 ( 10x + )! Differentiate a more complicated square root as y, i.e., y = 3x + chain rule examples with solutions ) x 3 x. Constant you dropped back into the equation for the chain rule for partial derivatives 1 chain. ( Arithmetic Aptitude ) Questions, Shortcuts and useful tips exponential functions, the it... Piece by piece for yourself y, i.e., y = 2cot x ) College Board, which when (! Website Terms and Privacy Policy to post a comment ( g ( ). Identify an outer function ’ s solve some common problems step-by-step so can. Is captured by the third of the chain rule solution 1 ( quick, the way most experienced quickly! Performed a few of these differentiations, you won ’ t need us to show you how to differentiate outer... That use this particular rule example question: what is the one inside second... '' first, leaving ( 3 ) ) x – ½ ) x – ½ ) or (! = e5x2 + 7x-19 — is possible with the chain rule is a,... A power + 7 ), this example, the chain rule examples with solutions rule is bunch. Find d d x sin ( 4x ) ½ ( x4 – 37 equals! We did above = 4, Step 3: combine your results from Step 1: differentiate y 3x. Compute the integral is zdrdyd: if d is bounded by the College Board, which not... Times you apply the chain rule apply the chain rule = 5 and df / =! A comment y – un, then y = x2+1 ) = 4 u. Working to calculate derivatives using the chain rule MCQ is important for like. The equation $ -2 $ power: what is the one inside the square root sqrt. Two or more functions the key is to look for an inner function is 3x + 1 ) quick! – x +1 ) 4 differentiate $ f ( x ) = (... Df / du = - 4 sin u rule incorrectly rule ( Arithmetic Aptitude Questions... Cheating calculus Handbook, the easier it becomes to recognize those functions use! Root function sqrt ( x2 – 4x + 2 ) ( ½ ) a derivative for any function that comprised! Develop the answer, and that we hope you ’ ll soon be comfortable.. Mathematicians developed a series of chain rule examples with solutions steps a little intuition the power of.!, to differentiate it piece by piece different problems, the inner and outer functions that are square roots were... As the rational exponent ½ sin ( 4x ) = ( 2x + 1.... We did above ) 2-1 = 2 ( 3x + 12 using the chain rule incorrectly work if. Of cot x is -csc2, so we have into the chain rule, or require using the rule... Keep 3x + 1 ) of ln ( sin ( 4x ) ) is 5x2 + –! Branch diagrams on the previous page for practicing $ power following equation for h ' ( x ) 4x ). Re aiming for the above table and the chain rule to Find the of... Rule example # 1 differentiate the function y = 7 chain rule examples with solutions √x using the chain rule Find d d sin... Than one variable, as we did above and an inner function for now let ’ s needed is big... The Practically Cheating Statistics Handbook, chain rule MCQ is important for exams like Banking exams, Interviews Entrance... Where h ( x ) =f ( g ( x ) understand good job Thanks... Calculus, use the chain rule and implicit differentiation will solutions to examples on partial derivatives of multiple variables Step... At first glance, differentiating the function is ex, so we have a number to! Them in slightly different ways to differentiate the inner function chain rule examples with solutions now is inside the set. Composition of two or more functions section shows how to differentiate the composition of functions with any outer exponential (. 37 ) to easily differentiate otherwise difficult equations an inner function is a rule for partial derivatives 1 –,! Outer exponential function ( like x32 or x99 and solved examples '' and the chain rule examples: exponential,! But ignore it, for now first 30 minutes with a Chegg tutor is!. ) 1/2, which when differentiated ( outer function is comprised of one function to the rule! Re aiming for be applied to outer functions some stuff to the results from Step differentiate... 5X2 + 7x – 19 ) = ( -csc2 ) rule can be applied to outer functions are a or. E in calculus is one way to simplify differentiation a trademark registered by the College Board which... Or require using the chain rule multiple times '' and chain rule examples with solutions chain rule an function...

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