# Waves/Derivatives

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### Math Tutorial -- Derivatives

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:
    ${\frac {dy}{dx}}\approx {\frac {\Delta y}{\Delta x}}.$ (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.
Table of Derivatives :
| ${d \over dx}c=0$ |
| ${d \over dx}x=1$ |
| ${d \over dx}cx=c$ |
| ${d \over dx}|x|={x \over |x|}=\operatorname {sgn} x,\qquad x\neq 0$ |
| ${d \over dx}x^{c}=cx^{c-1}$ | where both x$^c$ and cx$^{c−1}$ are defined. |
| ${d \over dx}\left({1 \over x}\right)={d \over dx}\left(x^{-1}\right)=-x^{-2}=-{1 \over x^{2}}$ |
| ${d \over dx}\left({1 \over x^{c}}\right)={d \over dx}\left(x^{-c}\right)=-{c \over x^{c+1}}$ |
| ${d \over dx}{\sqrt {x}}={d \over dx}x^{1 \over 2}={1 \over 2}x^{-{1 \over 2}}={1 \over 2{\sqrt {x}}}$ | x > 0 |
| ${d \over dx}c^{x}={c^{x}\ln c}$ | c > 0 |
| ${d \over dx}e^{x}=e^{x}$ |
| ${d \over dx}\log _{c}x={1 \over x\ln c}$ | c > 0, c ≠ 1 |
| ${d \over dx}\ln x={1 \over x}$ |
| ${d \over dx}\sin x=\cos x$ |
| ${d \over dx}\cos x=-\sin x$ |
| ${d \over dx}\tan x=\sec ^{2}x$ |
| ${d \over dx}\sec x=\tan x\sec x$ |
| ${d \over dx}\cot x=-\csc ^{2}x$ |
| ${d \over dx}\csc x=-\csc x\cot x$ |
| ${d \over dx}\arcsin x={1 \over {\sqrt {1-x^{2}}}}$ |
| ${d \over dx}\arccos x={-1 \over {\sqrt {1-x^{2}}}}$ |
| ${d \over dx}\arctan x={1 \over 1+x^{2}}$ |
| ${d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}$ |
| ${d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2}}$ |
| ${d \over dx}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1}}}$ |
| ${d \over dx}\sinh x=\cosh x$ |
| ${d \over dx}\cosh x=\sinh x$ |
| ${d \over dx}\tanh x=\operatorname {sech} ^{2}x$ |
| ${d \over dx}\operatorname {sech} x=-\tanh x\operatorname {sech} x$ |
| ${d \over dx}\operatorname {coth} x=-\operatorname {csch} ^{2}x$ |
| ${d \over dx}\operatorname {csch} x=-\operatorname {coth} x\operatorname {csch} x$ |
| ${d \over dx}\operatorname {arsinh} x={1 \over {\sqrt {x^{2}+1}}}$ |
| ${d \over dx}\operatorname {arcosh} x={1 \over {\sqrt {x^{2}-1}}}$ |
| ${d \over dx}\operatorname {artanh} x={1 \over 1-x^{2}}$ |
| ${d \over dx}\operatorname {arsech} x={1 \over x{\sqrt {1-x^{2}}}}$ |
| ${d \over dx}\operatorname {arcoth} x={1 \over 1-x^{2}}$ |
| ${d \over dx}\operatorname {arcsch} x={-1 \over |x|{\sqrt {1+x^{2}}}}$ |
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
    ${\frac {d}{dx}}\log(\sin(x))={\frac {1}{\sin(x)}}{\frac {d\sin(x)}{dx}}={\frac {\cos(x)}{\sin(x)}}.$
Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology