Calculus Course/Differentiation

Derivative

A derivative is a mathematical operation to find the rate of change of a function.

--174.117.102.136 (discuss) 20:19, 2 March 2012 (UTC)

Formula

For a non linear function f(x) = y . The rate of change of f(x) correspond to change of x is equal to the ratio of change in f(x) over change in x

$\frac{\Delta f(x)}{\Delta x} = \frac{\Delta y}{\Delta x}$

Then the Derivative of the function is defined as

$\frac{d}{dx} f(x) = Lim_{\Delta x \to 0} \Sigma \frac{\Delta f(x)}{\Delta x} = Lim_{\Delta x \to 0} \Sigma \frac{y}{x}$

but the derivative must exist uniquely at the point x. Seemingly well-behaved functions might not have derivatives at certain points. As examples, f(x)= 1/x has no derivative at x = 0; F(x) = |x| has two possible results at x = 0 (-1 for any value for which x<0 and +1 for any value for which x>0) On the other side, a function might have no value at x but a derivative of x, for example f(x)= x/x at x = 0. The function is undefined at x = 0, but the derivative is 0 at x = 0 as for any other value of x.

Practically all rules result, directly or indirectly, from a generalized treatment of the function.

Table of Derivative

General Rules

$\frac{d}{dx}(f + g)= \frac{df}{dx} + \frac{dg}{dx}$

$\frac{d}{dx}(cf)= c\frac{df}{dx}$

$\frac{d}{dx}(fg)= f\frac{dg}{dx} + g\frac{df}{dx}$

$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{g\frac{df}{dx} - f\frac{dg}{dx}}{g^2}$

Powers and Polynomials

• $\frac{d}{dx} (c) = 0$
• $\frac{d}{dx}x=1$
• $\frac{d}{dx}x^n=nx^{n-1}$
• $\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt x}$
• $\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$
• $\frac{d}{dx}(c_n x^n + c_{n-1} x^{n-1} + c_{n-2}x^{n-2} + \cdots +c_2x^2 + c_1 x + c_0) = n c_n x^{n-1} + (n-1) c_{n-1} x^{n-2} + (n-2) c_{n-2}x^{n-3} + \cdots + 2c_2x+ c_1$

Trigonometric Functions

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

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

$\frac{d}{dx} \tan (x)= \sec^2 (x)$

$\frac{d}{dx} \cot (x)= -\csc^2 (x)$

$\frac{d}{dx} \sec (x)= \sec (x) \tan (x)$

$\frac{d}{dx} \csc (x) = -\csc (x) \cot (x)$

Exponential and Logarithmic Functions

• $\frac{d}{dx} e^x =e^x$
• $\frac{d}{dx} a^x =a^x \ln (a)\qquad\mbox{if }a>0$
• $\frac{d}{dx} \ln (x)= \frac{1}{x}$
• $\frac{d}{dx} \log_a (x)= \frac{1}{x\ln (a)}\qquad\mbox{if }a>0, a\neq 1$
• $(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\qquad f > 0$
• $(c^f)' = \left(e^{f\ln c}\right)' = f' c^f \ln c$

Inverse Trigonometric Functions

• $\frac{d}{dx} \mbox{arcsin x} = \frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx} \mbox{arccos x} = -\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx} \mbox{arctan x} = \frac{1}{1+x^2}$
• ${d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}$
• ${d \over dx} \arccot x = {-1 \over 1 + x^2}$
• ${d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}$

Hyperbolic and Inverse Hyperbolic Functions

${d \over dx} \sinh x = \cosh x$

${d \over dx} \cosh x = \sinh x$

${d \over dx} \tanh x = \mbox{sech}^2\,x$

${d \over dx} \,\mbox{sech}\,x = -\tanh x\,\mbox{sech}\,x$

${d \over dx} \,\mbox{coth}\,x = -\,\mbox{csch}^2\,x$

${d \over dx} \,\mbox{csch}\,x = -\,\mbox{coth}\,x\,\mbox{csch}\,x$

${d \over dx} \sinh^{-1} x = { 1 \over \sqrt{x^2 + 1}}$

${d \over dx} \cosh^{-1} x = {-1 \over \sqrt{x^2 - 1}}$

${d \over dx} \tanh^{-1} x = { 1 \over 1 - x^2}$

${d \over dx} \mbox{sech}^{-1}\,x = { 1 \over x\sqrt{1 - x^2}}$

${d \over dx} \mbox{coth}^{-1}\,x = {-1 \over 1 - x^2}$

${d \over dx} \mbox{csch}^{-1}\,x = {-1 \over |x|\sqrt{1 + x^2}}$

Reference

1. Derivative
2. Table_of_derivatives