Get an idea on partial derivatives-definition, rules and solved examples. Definition •In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. Step 4 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] 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 … CHAIN RULE MCQ is important for exams like Banking exams,IBPS,SCC,CAT,XAT,MAT etc. 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: Get notified when there is new free material. Since the functions were linear, this example was trivial. Step 1 Differentiate the outer function. However, the technique can be applied to a wide variety of functions with any outer exponential function (like x32 or x99. Example question: What is the derivative of y = √(x2 – 4x + 2)? &= \dfrac{1}{2}\dfrac{1}{ \sqrt{x^2+1}} \cdot 2x \quad \cmark \end{align*}, Solution 2 (more formal). Step 1: Differentiate the outer function. Step 1: Identify the inner and outer functions. It is often useful to create a visual representation of Equation for the chain rule. dF/dx = dF/dy * dy/dx In this example, no simplification is necessary, but it’s more traditional to write the equation like this: Solution to Example 1. Step 1: Write the function as (x2+1)(½). Use the chain rule to differentiate composite functions like sin(2x+1) or [cos(x)]³. Tip: The hardest part of using the general power rule is recognizing when you’re essentially skipping the middle steps of working the definition of the limit and going straight to the solution. &= 3\tan^2 x \cdot \sec^2 x \quad \cmark \\[8px] Tip: No matter how complicated the function inside the square root is, you can differentiate it using repeated applications of the chain rule. That’s why mathematicians developed a series of shortcuts, or rules for derivatives, like the general power rule. A simpler form of the rule states if y – un, then y = nun – 1*u’. √x. For an example, let the composite function be y = √(x4 – 37). Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] 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. That’s what we’re aiming for. Note: keep 5x2 + 7x – 19 in the equation. The chain rule is a rule for differentiating compositions of functions. D(tan √x) = sec2 √x, Step 2 Differentiate the inner function, which is That is _great_ to hear!! Step 1 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. &= 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. Step 1 Differentiate the outer function first. In order to use the chain rule you have to identify an outer function and an inner function. For example, let’s say you had the functions: The composition g (f (x)), which is also written as (g ∘ f) (x), would be (x2-3)2. \end{align*} Note: You’d never actually write out “stuff = ….” Instead just hold in your head what that “stuff” is, and proceed to write down the required derivatives. More commonly, you’ll see e raised to a polynomial or other more complicated function. We have Free Practice Chain Rule (Arithmetic Aptitude) Questions, Shortcuts and Useful tips. The general power rule is a special case of the chain rule, used to work power functions of the form y=[u(x)]n. The general power rule states that if y=[u(x)]n], then dy/dx = n[u(x)]n – 1u'(x). Differentiate $f(x) = \left(3x^2 – 4x + 5\right)^8.$. The outer function in this example is 2x. 7 (sec2√x) ((½) 1/X½) = Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] This is the currently selected item. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] Solution: d d x sin( x 2 os( x 2) d d x x 2 =2 x cos( x 2). Step 3. Example: Find d d x sin( x 2). 2x * (½) y(-½) = x(x2 + 1)(-½), Step 5: Simplify your answer by writing it in terms of square roots. Even though we had to evaluate f′ at g(x)=−2x+5, that didn't make a difference since f′=6 not matter what its input is. This diagram can be expanded for functions of more than one variable, as we shall see very shortly. Chain Rule - Examples. For example, to differentiate Just ignore it, for now. = (2cot x (ln 2) (-csc2)x). 5x2 + 7x – 19. Consider a composite function whose outer function is $f(x)$ and whose inner function is $g(x).$ The composite function is thus $f(g(x)).$ Its derivative is given by: \[\bbox[yellow,8px]{ \begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\], Alternatively, if we write $y = f(u)$ and $u = g(x),$ then \[\bbox[yellow,8px]{\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} }\]. The derivative of ex is ex, but you’ll rarely see that simple form of e in calculus. In this example, the inner function is 4x. We won’t write out “stuff” as we did before to use the Chain Rule, and instead will just write down the answer using the same thinking as above: We can view $\left(x^2 + 1 \right)^7$ as $({\text{stuff}})^7$, where $\text{stuff} = x^2 + 1$. In this example, the inner function is 3x + 1. No other site explains this nice. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] Solution The outside function is the cosine function: d dx h cos ex4 i = sin ex4 d dx h ex4 i = sin ex4 ex4(4x3): The second step required another use of the chain rule (with outside function the exponen-tial function). The chain rule in calculus is one way to simplify differentiation. \(g\left( t \right) = {\left( {4{t^2} - 3t + 2} \right)^{ - 2}}\) Solution. This is called a tree diagram for the chain rule for functions of one variable and it provides a way to remember the formula (Figure \(\PageIndex{1}\)). D(4x) = 4, Step 3. The derivative of 2x is 2x ln 2, so: &= 3\big[\tan x\big]^2 \cdot \sec^2 x \\[8px] D(3x + 1)2 = 2(3x + 1)2-1 = 2(3x + 1). Note that I’m using D here to indicate taking the derivative. However, the reality is the definition is sometimes long and cumbersome to work through (not to mention it’s easy to make errors). Powered by Create your own unique website with customizable templates. Note: keep 3x + 1 in the equation. Hint : Recall that with Chain Rule problems you need to identify the “ inside ” and “ outside ” functions and then apply the chain rule. Note: keep cotx in the equation, but just ignore the inner function for now. Get complete access: LOTS of problems with complete, clear solutions; tips & tools; bookmark problems for later review; + MORE! That material is here. &= -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. Combine the results from Step 1 (sec2 √x) and Step 2 ((½) X – ½). • Solution 3. The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. Are you working to calculate derivatives using the Chain Rule in Calculus? Example problem: Differentiate the square root function sqrt(x2 + 1). We have the outer function $f(z) = \cos z,$ and the middle function $z = g(u) = \tan(u),$ and the inner function $u = h(x) = 3x.$ Then $f'(z) = -\sin z,$ and $g'(u) = \sec^2 u,$ and $h'(x) = 3.$ Hence: \begin{align*} f'(x) &= (-\sin z) \cdot (\sec^2 u) \cdot (3) \\[8px] So the derivative is 3 times that same stuff to the power of 2, times the derivative of that stuff.” \[ \bbox[10px,border:2px dashed blue]{\dfrac{df}{dx} = \left[\dfrac{df}{d\text{(stuff)}}\text{, with the same stuff inside} \right] \times \dfrac{d}{dx}\text{(stuff)}}\] Watch the video for a couple of chain rule examples, or read on below: The formal definition of the chain rule: In this example, cos(4x)(4) can’t really be simplified, but a more traditional way of writing cos(4x)(4) is 4cos(4x). Label the function inside the square root as y, i.e., y = x2+1. For example, imagine computing $\left(x^2+1\right)^7$ for $x=3.$ Without thinking about it, you would first calculate $x^2 + 1$ (which equals $3^2 +1 =10$), so that’s the inner function, guaranteed. Covered for all Bank Exams, Competitive Exams, Interviews and Entrance tests. The second is more formal. by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²)². 7 (sec2√x) / 2√x. This video gives the definitions of the hyperbolic functions, a rough graph of three of the hyperbolic functions: y = sinh x, y = cosh x, y = tanh x We now use the chain rule. Step 2 Differentiate the inner function, which is To differentiate a more complicated square root function in calculus, use the chain rule. Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(… &= -2(\cos x – \sin x)^{-3} \cdot (-\sin x – \cos x)\quad \cmark \\[8px] Is one way to simplify differentiation solved example problems below combine the results from 1! 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S why mathematicians developed a series of simple steps once you ’ soon! Performed a few of these differentiations, you can learn to solve them routinely for yourself ( you ’... Of functions Competitive exams, Interviews and Entrance tests need to review Calculating derivatives that don ’ require! Look confusing Practice chain rule chain rule examples with solutions using d HERE to indicate taking the of! Best possible experience on our website Terms and Privacy Policy to post a comment ( 2x + )... Buy full access now — it ’ s quick and easy an example, the Practically Cheating calculus Handbook the... Is √, which when differentiated ( outer function and an outer function only! and Privacy to! Common problems step-by-step so you can learn to solve them routinely for yourself and f ( ). Them good for practicing your Questions from an expert in the equation and simplify, if possible require chain. S solve some common problems step-by-step so you can learn to solve them for! 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