Calculus Course/Differentiation

Derivative

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

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}\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 Derivative

General Rules

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

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

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

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

Powers and Polynomials

${\frac {d}{dx}}(c)=0$

${\frac {d}{dx}}x=1$

${\frac {d}{dx}}x^{n}=nx^{n-1}$

${\frac {d}{dx}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}$

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

${{\frac {d}{dx}}(c_{n}x^{n}+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_{2}x^{2}+c_{1}x+c_{0})=nc_{n}x^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_{2}x+c_{1}}$

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

${\frac {d}{dx}}e^{x}=e^{x}$

${\frac {d}{dx}}a^{x}=a^{x}\ln(a)\qquad {\mbox{if }}a>0$

${\frac {d}{dx}}\ln(x)={\frac {1}{x}}$

${\frac {d}{dx}}\log _{a}(x)={\frac {1}{\ln(a)x}}\qquad {\mbox{if }}a>0,a\neq 1$

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

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

Inverse Trigonometric Functions

${\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}$

${\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}$

${\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}$

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

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

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

Hyperbolic and Inverse Hyperbolic Functions

{\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}}}}}

Reference

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