1.8: Math Review — Derivatives
- Page ID
- 33144
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This section provides a quick review of the idea of the derivative. Often we are interested in the slope of a line tangent to a function y(x) at some value of x. This slope is called the derivative and is denoted dy∕dx. Since a tangent line to the function can be defined at any point x, the derivative itself is a function of x:
\[g(x)=\frac{d y(x)}{d x}\label{1.25}\]
As Figure \(\PageIndex{15}\): illustrates, the slope of the tangent line at some point on the function may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve:
\[\frac{d y}{d x} \approx \frac{\Delta y}{\Delta x}\label{1.26}\]
As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact.
Derivatives of some common functions are now given. In each case a is a constant.
\[\frac{d x^{a}}{d x}=a x^{a-1}\label{1.27}\]
\[\frac{d}{d x} \exp (a x)=a \exp (a x)\label{1.28}\]
\[\frac{d}{d x} \log (a x)=\frac{1}{x}\label{1.29}\]
\[\frac{d}{d x} \sin (a x)=a \cos (a x)\label{1.30}\]
\[\frac{d}{d x} \cos (a x)=-a \sin (a x)\label{1.31}\]
\[\frac{d a f(x)}{d x}=a \frac{d f(x)}{d x}\label{1.32}\]
\[\frac{d}{d x}[f(x)+g(x)]=\frac{d f(x)}{d x}+\frac{d g(x)}{d x}\label{1.33}\]
\[\frac{d}{d x} f(x) g(x)=\frac{d f(x)}{d x} g(x)+f(x) \frac{d g(x)}{d x} \quad \text { (product rule) }\label{1.34}\]
\[\frac{d}{d x} f(y)=\frac{d f}{d y} \frac{d y}{d x} \quad(\text { chain rule })\label{1.35}\]
The product and chain rules are used to compute the derivatives of complex functions. For instance,
\(\frac{d}{d x}(\sin (x) \cos (x))=\frac{d \sin (x)}{d x} \cos (x)+\sin (x) \frac{d \cos (x)}{d x}=\cos ^{2}(x)-\sin ^{2}(x)\)
and
\(\frac{d}{d x} \log (\sin (x))=\frac{1}{\sin (x)} \frac{d \sin (x)}{d x}=\frac{\cos (x)}{\sin (x)}\).