Calculus Course/Differentiation

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Contents

DerivativeEdit

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

FormulaEdit

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}\sum\frac{\Delta f(x)}{\Delta x}=\lim_{\Delta x\to 0}\sum\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)=\frac{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)=\frac{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 DerivativeEdit

General RulesEdit

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

\frac{d}{dx}(c\cdot f)=c\cdot\frac{df}{dx}

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

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

Powers and PolynomialsEdit

\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_nx^n+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots+c_2x^2+c_1x+c_0)=nc_nx^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots+2c_2x+c_1}

Trigonometric FunctionsEdit

\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 FunctionsEdit

\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}{\ln(a)x}\qquad\mbox{if }a>0, a\ne 1

\frac{d}{dx}(f^g)=\frac{d}{dx}\left(e^{g\ln(f)}\right) = f^g\left(\frac{df}{dx}\cdot\frac{g}{f}+\frac{dg}{dx}\cdot\ln(f)\right),\qquad f>0

\frac{d}{dx}(c^f)=\frac{d}{dx}\left(e^{f\ln(c)}\right)=\frac{df}{dx}\cdot c^f\ln(c)

Inverse Trigonometric FunctionsEdit

\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}\arcsec(x)=\frac{1}{|x|\sqrt{x^2-1}}

\frac{d}{dx}\arccot(x)=-\frac{1}{1+x^2}

\frac{d}{dx}\arccsc(x)=-\frac{1}{|x|\sqrt{x^2-1}}

Hyperbolic and Inverse Hyperbolic FunctionsEdit

\frac{d}{dx}\sinh(x)=\cosh(x)
\frac{d}{dx}\cosh(x)=\sinh(x)
\frac{d}{dx}\tanh(x)={\rm sech}^2(x)
\frac{d}{dx}{\rm sech}(x)=-\tanh(x){\rm sech}(x)
\frac{d}{dx}\coth(x)=-{\rm csch}^2(x)
\frac{d}{dx}{\rm csch}(x)=-\coth(x){\rm csch}(x)
\frac{d}{dx}{\rm arcsinh}(x)=\frac{1}{\sqrt{x^2+1}}
\frac{d}{dx}{\rm arccosh}(x)=-\frac{1}{\sqrt{x^2-1}}
\frac{d}{dx}{\rm arctanh}(x)=\frac{1}{1-x^2}
\frac{d}{dx}{\rm arcsech}(x)=\frac{1}{x\sqrt{1-x^2}}
\frac{d}{dx}{\rm arccoth}(x)=-\frac{1}{1-x^2}
\frac{d}{dx}{\rm arccsch}(x)=-\frac{1}{|x|\sqrt{1+x^2}}

ReferenceEdit