Waves/Derivatives

Math Tutorial -- Derivatives

Figure 1.15: Estimation of the derivative, which is the slope of the tangent line. When point B approaches point A, the slope of the line AB approaches the slope of the tangent to the curve at point A.

This section provides a quick introduction to the idea of the derivative. For a more detailed discussion and exploration of the differentiation and of Calculus, see Calculus and Differentiation.

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 $x$:

$g(x)=\frac{dy(x)}{dx}.$ (2.25)

As figure 1.15 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:

$\frac{dy}{dx}\approx \frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}.$ (2.26)

As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact.

Table of Derivatives

Derivatives of some common functions are now given. In each case $c$ is a constant.

Template:Wikipedia

Table of Derivatives
$\frac{d}{dx}c=0$
$\frac{d}{dx}x=1$
$\frac{d}{dx}cx=c$
$\frac{d}{dx}|x|=\frac{x}{|x|}=\mathrm{sgn}x,x\ne 0$
$\frac{d}{dx}{x}^c=c{x}^{c-1}$	where both xc and cxc−1 are defined.
$\frac{d}{dx}\left(\frac{1}{x}\right)=\frac{d}{dx}\left(x^{-1}\right)=-x^{-2}=-\frac{1}{x^2}$
$\frac{d}{dx}\left(\frac{1}{x^c}\right)=\frac{d}{dx}\left(x^{-c}\right)=-\frac{c}{x^{c+1}}$
$\frac{d}{dx}\sqrt{x}=\frac{d}{dx}{x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$	x > 0
$\frac{d}{dx}{c}^x=c^x\ln c$	c > 0
$\frac{d}{dx}{e}^x=e^x$
$\frac{d}{dx}{\log }_{c}x=\frac{1}{x\ln c}$	c > 0, c ≠ 1
$\frac{d}{dx}\ln x=\frac{1}{x}$
$\frac{d}{dx}\sin x=\cos x$
$\frac{d}{dx}\cos x=-\sin x$
$\frac{d}{dx}\tan x={\sec }^{2}x$
$\frac{d}{dx}\sec x=\tan x\sec x$
$\frac{d}{dx}\cot x=-{\csc }^{2}x$
$\frac{d}{dx}\csc x=-\csc x\cot x$
$\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\arccos x=\frac{-1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$
$\frac{d}{dx}\mathrm{arcsec}x=\frac{1}{|x|\sqrt{x^2-1}}$
$\frac{d}{dx}\mathrm{arccot}x=\frac{-1}{1+x^2}$
$\frac{d}{dx}\mathrm{arccsc}x=\frac{-1}{|x|\sqrt{x^2-1}}$
$\frac{d}{dx}\sinh x=\cosh x$
$\frac{d}{dx}\cosh x=\sinh x$
$\frac{d}{dx}\tanh x={\mathrm{sech}}^{2}x$
$\frac{d}{dx}\mathrm{sech}x=-\tanh x\mathrm{sech}x$
$\frac{d}{dx}\coth x=-{\mathrm{csch}}^{2}x$
$\frac{d}{dx}\mathrm{csch}x=-\coth x\mathrm{csch}x$
$\frac{d}{dx}\mathrm{arsinh}x=\frac{1}{\sqrt{x^2+1}}$
$\frac{d}{dx}\mathrm{arcosh}x=\frac{1}{\sqrt{x^2-1}}$
$\frac{d}{dx}\mathrm{artanh}x=\frac{1}{1-x^2}$
$\frac{d}{dx}\mathrm{arsech}x=\frac{1}{x\sqrt{1-x^2}}$
$\frac{d}{dx}\mathrm{arcoth}x=\frac{1}{1-x^2}$
$\frac{d}{dx}\mathrm{arcsch}x=\frac{-1}{|x|\sqrt{1+x^2}}$

The product and chain rules are used to compute the derivatives of complex functions. For instance,

$\frac{d}{dx}(\sin (x)\cos (x))=\frac{d\sin (x)}{dx}\cos (x)+\sin (x)\frac{d\cos (x)}{dx}={\cos }^2(x)-{\sin }^2(x)$

and

$\frac{d}{dx}\log (\sin (x))=\frac{1}{\sin (x)}\frac{d\sin (x)}{dx}=\frac{\cos (x)}{\sin (x)}.$

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