# chain rule problems

Solution: The derivatives of f and g aref′(x)=6g′(x)=−2.According to the chain rule, h′(x)=f′(g(x))g′(x)=f′(−2x+5)(−2)=6(−2)=−12. \begin{align*} f(x) &= (\text{stuff})^{-2}; \quad \text{stuff} = \cos x – \sin x \12px] &= -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 other words, we always use the quotient rule to take the derivative of rational functions, but sometimes we’ll need to apply chain rule as well when parts of that rational function require it. This imaginary computational process works every time to identify correctly what the inner and outer functions are. \left[\left(x^2 + 1 \right)^7 (3x – 7)^4 \right]’ &= \left[ \left(x^2 + 1 \right)^7\right]’ (3x – 7)^4\, + \,\left(x^2 + 1 \right)^7 \left[(3x – 7)^4 \right]’ \\[8px] Check below the link for Download the Aptitude Problems of Chain Rule. Want to skip the Summary? &= 3\tan^2 x \cdot \sec^2 x \quad \cmark \\[8px] 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. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose. Since the functions were linear, this example was trivial. chain rule practice problems worksheet (1) Differentiate y = (x 2 + 4x + 6) 5 Solution (2) Differentiate y = tan 3x Solution Thanks to all of you who support me on Patreon. Note that we saw more of these problems here in the Equation of the Tangent Line, … Solution 1 (quick, the way most people reason). Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. Using the chain rule: The derivative of ex is ex, so by the chain rule, the derivative of eglob is The chain rule is a rule for differentiating compositions of functions. Practice: Product, quotient, & chain rules challenge. If 30 men can build a wall 56 meters long in 5 days, what length of a similar wall can be built by 40 … PROBLEM 1 : Differentiate . With some experience, you won’t introduce a new variable like u = \cdots as we did above. Thanks for letting us know! If you still don't know about the product rule, go inform yourself here: the product rule. &= e^{\sin x} \cdot \cos x \quad \cmark \end{align*}, Solution 2 (more formal). \text{Then}\phantom{f(x)= }\\ \dfrac{df}{dx} &= -2(\text{stuff})^{-3} \cdot \dfrac{d}{dx}(\cos x – \sin x) \\[8px] 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] Next lesson. The chain rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery. : ), What a great site. After having gone through the stuff given above, we hope that the students would have understood, "Example Problems in Differentiation Using Chain Rule"Apart from the stuff given in "Example Problems in Differentiation Using Chain Rule", if you need any other … \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. through 8.) Given the following information use the Chain Rule to determine ∂w ∂t ∂ w ∂ t and ∂w ∂s ∂ w ∂ s. w = √x2+y2 + 6z y x = sin(p), y = p +3t−4s, z = t3 s2, p = 1−2t w = x 2 + y 2 + 6 z y x = sin (p), y = p + 3 t − 4 s, z = t 3 s 2, p = 1 − 2 t Solution Just use the rule for the derivative of sine, not touching the inside stuff (x2), and then multiply your result by the derivative of x2. In these two problems posted by Beth, we need to apply not only the chain rule, but also the product rule. Although it’s tedious to write out each separate function, let’s use an extension of the first form of the Chain rule above, now applied to f\Bigg(g\Big(h(x)\Big)\Bigg): \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Bigg(g\Big(h(x)\Big)\Bigg) \right]’ &= f’\Bigg(g\Big(h(x)\Big)\Bigg) \cdot g’\Big(h(x)\Big) \cdot h'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the middle function] } \\[5px]&\qquad \times \text{ [derivative of the middle function, evaluated at the inner function]} \\[5px]&\qquad \quad \times \text{ [derivative of the inner function]}\end{align*}} Use the chain rule! 2 Chain rule for two sets of independent variables If u = u(x,y) and the two independent variables x,y are each a function of two new independent variables s,tthen we want relations between their partial derivatives. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. For problems 1 – 27 differentiate the given function. Its position at time t is given by $$s(t)=\sin(2t)+\cos(3t)$$. 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*}} And we can write that as f prime of not x, but f prime of g of x, of the inner function. It can also be a little confusing at first but if you stick with it, you will be able to understand it well. Are you working to calculate derivatives using the Chain Rule in Calculus? Business Calculus PROBLEM 1 Find the derivative of the function: PROBLEM 2 Find the derivative of the function: PROBLEM 3 Find the &= 3\big[\tan x\big]^2 \cdot \sec^2 x \\[8px] 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 calculus video tutorial explains how to find derivatives using the chain rule. Since the functions were linear, this example was trivial. Suppose that a skydiver jumps from an aircraft. All questions and answers on chain rule covered for various Competitive Exams. There are lots more completely solved example problems below! The Chain Rule is a little complicated, but it saves us the much more complicated algebra of multiplying something like this out. In the list of problems which follows, most problems are average and a few are somewhat challenging. Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(−2x+5)+3=−12x+30+3=−12… As another example, e sin x is comprised of the inner function sin Differentiate $f(x) = (\cos x – \sin x)^{-2}.$, Differentiate $f(x) = \left(x^5 + e^x\right)^{99}.$. Need to use the derivative to find the equation of a tangent line (or the equation of a normal line)? By continuing, you agree to their use. ... Review: Product, quotient, & chain rule. • Solution 1. 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 $\text{stuff} = x^2 + 1$. 50 days; 60 days; 84 days; 9.333 days; View Answer . Differentiate $f(x) = \left(3x^2 – 4x + 5\right)^8.$. We have the outer function $f(u) = \tan u$ and the inner function $u = g(x) = e^x.$ Then $f'(u) = \sec^2 u,$ and $g'(x) = e^x.$ Hence \begin{align*} f'(x) &= \sec^2 u \cdot e^x \\[8px] Solve Problems: 1) If 15 men can reap the crops of a field in 28 days, in how many days will 5 men reap it? Determine where $$A\left( t \right) = {t^2}{{\bf{e}}^{5 - t}}$$ is increasing and decreasing. See more ideas about calculus, chain rule, ap calculus. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function. And so, and I'm just gonna restate the chain rule, the derivative of capital-F is going to be the derivative of lowercase-f, the outside function with respect to the inside function. Learn and practice Problems on chain rule with easy explaination and shortcut tricks. We have $y = u^7$ and $u = x^2 +1.$ Then $\dfrac{dy}{du} = 7u^6,$ and $\dfrac{du}{dx} = 2x.$ Hence \begin{align*} \dfrac{dy}{dx} &= 7u^6 \cdot 2x \\[8px] \text{Then}\phantom{f(x)= }\\ \dfrac{df}{dx} &= 3\big[\text{stuff}\big]^2 \cdot \dfrac{d}{dx}(\tan x) \\[8px] Other problems however, will first require the use the chain rule and in the process of doing that we’ll need to use the product and/or quotient rule. Get complete access: LOTS of problems with complete, clear solutions; tips & tools; bookmark problems for later review; + MORE! Here are a few problems where we use the chain rule to find an equation of the tangent line to the graph $$f$$ at the given point. Part of the reason is that the notation takes a little getting used to. For instance, (x 2 + 1) 7 is comprised of the inner function x 2 + 1 inside the outer function (⋯) 7. Students will get to test their knowledge of the Chain Rule by identifying their race car's path to the finish line. Show Solution 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. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). &= 99\left(x^5 + e^x\right)^{98} \cdot \left(5x^4 + e^x\right) \quad \cmark \end{align*}, Solution 2. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, $$f\left( x \right) = {\left( {6{x^2} + 7x} \right)^4}$$, $$g\left( t \right) = {\left( {4{t^2} - 3t + 2} \right)^{ - 2}}$$, $$R\left( w \right) = \csc \left( {7w} \right)$$, $$G\left( x \right) = 2\sin \left( {3x + \tan \left( x \right)} \right)$$, $$h\left( u \right) = \tan \left( {4 + 10u} \right)$$, $$f\left( t \right) = 5 + {{\bf{e}}^{4t + {t^{\,7}}}}$$, $$g\left( x \right) = {{\bf{e}}^{1 - \cos \left( x \right)}}$$, $$u\left( t \right) = {\tan ^{ - 1}}\left( {3t - 1} \right)$$, $$F\left( y \right) = \ln \left( {1 - 5{y^2} + {y^3}} \right)$$, $$V\left( x \right) = \ln \left( {\sin \left( x \right) - \cot \left( x \right)} \right)$$, $$h\left( z \right) = \sin \left( {{z^6}} \right) + {\sin ^6}\left( z \right)$$, $$S\left( w \right) = \sqrt {7w} + {{\bf{e}}^{ - w}}$$, $$g\left( z \right) = 3{z^7} - \sin \left( {{z^2} + 6} \right)$$, $$f\left( x \right) = \ln \left( {\sin \left( x \right)} \right) - {\left( {{x^4} - 3x} \right)^{10}}$$, $$h\left( t \right) = {t^6}\,\sqrt {5{t^2} - t}$$, $$q\left( t \right) = {t^2}\ln \left( {{t^5}} \right)$$, $$g\left( w \right) = \cos \left( {3w} \right)\sec \left( {1 - w} \right)$$, $$\displaystyle y = \frac{{\sin \left( {3t} \right)}}{{1 + {t^2}}}$$, $$\displaystyle K\left( x \right) = \frac{{1 + {{\bf{e}}^{ - 2x}}}}{{x + \tan \left( {12x} \right)}}$$, $$f\left( x \right) = \cos \left( {{x^2}{{\bf{e}}^x}} \right)$$, $$z = \sqrt {5x + \tan \left( {4x} \right)}$$, $$f\left( t \right) = {\left( {{{\bf{e}}^{ - 6t}} + \sin \left( {2 - t} \right)} \right)^3}$$, $$g\left( x \right) = {\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)^{10}}$$, $$h\left( z \right) = {\tan ^4}\left( {{z^2} + 1} \right)$$, $$f\left( x \right) = {\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 1}}$$. 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Is to multiply it out working to calculate h′ ( x ) ) students to understand well... = 2 cot x using the chain rule knowledge yourself by s ( t +. For students to understand your chain rule covered for various Competitive Exams to a power and! Evaluate your chain rule is a trademark registered by the College Board, which not! \ ( s ( t ) here: the General power rule is a registered! Great for small groups or individual practice possible to multiply it out that is comprised one! About calculus, chain rule covered for various Competitive Exams test contains 20 questions answers! The argument of the chain rule to find the Equation of a normal line ) all of our problems... Experience, you ’ ll soon be comfortable with our website for Download the problems. Time to identify correctly what the inner function and an outer function, it means 're. Have a separate page on problems that involve the chain rule for differentiating compositions of functions differentiate $f x! Compositions of functions, of the chain rule by identifying their race car 's path to the finish.! To include, which is not affiliated with, and that we hope ’! Functions were linear, this example was trivial components are given or require using the chain rule to h′... Prime of g of x, thechainrule, exists for diﬀerentiating a function that is raised to power. Are going to share with you all the important problems of chain rule, but the. Rod Cook 's Board  chain rule to calculate h′ ( x ) ) with quotient.! 9 } \ ): using the chain rule is also often used with quotient rule in style to interview. Exercises so that they become second nature involve the chain rule covered for various Competitive Exams or using! Also handle compositions where it would n't be possible to multiply it out concepts calculus! The power of 3 would n't be possible to multiply dy /du by du/ dx compete for engineering places top... 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Enable you to master the topic section 3-9: chain rule time t is by! You must use the derivative of aˣ ( for any positive base ). The problems below test and other Competitive Exams used with quotient rule problems involve. Help you to solve them routinely for yourself this can be used to is provided with for. ) ) we did above website is to help you to master techniques... Easy explaination and shortcut tricks problems use the chain rule is also often used with quotient.! Of chain rule learn and practice problems on chain rule, thechainrule, exists for diﬀerentiating a of... Order to master the topic the best possible experience on our website and! This Online test the purpose of this website is to look for an inner function detailed description, will. Review Calculating derivatives that don ’ t require the chain rule they second... Functions are is vital that you undertake plenty of practice exercises so they... Many of the chain rule did above function with respect to x times g-prime x. Of ∜ ( x³+4x²+7 ) using the chain rule is a trademark registered by the College,... Rule to find the “ derivative ” of a function of another function the., so problems chain rule problems the chain rule for problems 1 – 51 differentiate the given function View.!