2.6: The Chain Rule
- Page ID
- 204098
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.6 The Chain Rule
- Given a composite function, identify the inner and outer functions, and describe how they relate to the overall structure of the function.
- Use the chain rule in conjunction with the power, product, and quotient rules to calculate the derivative of composite functions, applying the appropriate rules in the correct order.
Let \( h(x) \) be the composite function \( h(x) = f(g(x)) \). The function \( g \) is applied first to the input \( x \), and is referred to as the inner function.
The function \( f \) is applied to the result \( g(x) \), and is referred to as the outer function.
For each composite function below, identify the inner and outer functions.
- \( h(x) = (x^2 + 5)^{10} \)
- \( h(x) = \sqrt{x^3 + 4x + 1} \)
- \( h(x) = \sin(x^7) \)
- \( h(x) = \cos^3(x) \)
The Chain Rule
The Chain Rule is a method to calculate the derivative of composite functions:
Suppose that \( g \) is differentiable at \( a \) and \( f \) is differentiable at \( g(a) \).
Then the composite function \( h = f \circ g \), defined by \( h(x) = f(g(x)) \), is differentiable at \( a \), and
\[
h'(a) = f'(g(a)) \cdot g'(a)
\]
Problem-solving strategy to find the derivative of \( h(x) = f(g(x)) \):
- Identify \( f(x) \) and \( g(x) \) in the composition \( h(x) = f(g(x)) \)
- Find \( f'(x) \), then evaluate it at \( g(x) \) to obtain \( f'(g(x)) \)
- Find \( g'(x) \)
- Write the derivative: \( h'(x) = f'(g(x)) \cdot g'(x) \)
Calculate the derivative of each function:
- \( h(x) = (x^2 + 5)^{10} \)
- \( h(x) = \sqrt{x^3 + 4x + 1} \)
- \( h(x) = \cos^3 x \)
- \( h(x) = \sin(x^7 + 4x^3) \)
- \( h(x) = \dfrac{1}{\sin^3 x} \)
- \( h(x) = (3x + 8)^4 (x^2 - 3x)^2 \)
- \( h(x) = \dfrac{3x + \tan x}{(6x^2 + 1)^5} \)
- \( h(x) = \cos^{31}(3x^5 + 2x) \)
We can also use the Leibniz notation to find the derivative of \( h(x) = f(g(x)) \).
If \( y = f(u) \) and \( u = g(x) \) are both differentiable, then:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]
Use the Leibniz notation to find the derivative of:
- \( y = (x^2 + 5)^{10} \)
- \( y = \sin(x^7 + 4x^3) \)
Find an equation of the tangent line to the graph of \( f(x) = \cos(\pi x) \) at the point \( \left( \frac{1}{2}, 0 \right) \).
The position function of a train is given by
\[
s(t) = \frac{200}{(1 + t)^3}
\]
where \( s \) is in meters and \( t \) is in seconds.
- Find the velocity and acceleration of the train at \( t = 6 \).
- Is the train speeding up or slowing down at that moment?