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Differentiation Rules

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Function to Differentiate: $s(x) = 1/x$

Derivative of the Function: $s'(x) = -\frac{1}{x^2}$

Function to Differentiate: $d(x) = \log_{a}(x)$ , where $a > 0$ and $a \neq 1$

Derivative of the Function: $d'(x) = \frac{1}{x\ln(a)}$

Function to Differentiate: $t(x) = \sqrt{x}$

Derivative of the Function: $t'(x) = \frac{1}{2\sqrt{x}}$

Function to Differentiate: $w(x) = \ln(\sin(x))$

Derivative of the Function: $w'(x) = \frac{\cos(x)}{\sin(x)}$

Function to Differentiate: $c(x) = \frac{x^2}{2} - \frac{1}{x}$

Derivative of the Function: $c'(x) = x + \frac{1}{x^2}$

Function to Differentiate: $f(x) = x^3$

Derivative of the Function: $f'(x) = 3x^2$

Function to Differentiate: $z(x) = e^{x^2}$

Derivative of the Function: $z'(x) = 2xe^{x^2}$

Function to Differentiate: $q(x) = \cos(x)$

Derivative of the Function: $q'(x) = -\sin(x)$

Function to Differentiate: $l(x) = \arccos(x)$

Derivative of the Function: $l'(x) = -\frac{1}{\sqrt{1 - x^2}}$

Function to Differentiate: $p(x) = \sin(x)$

Derivative of the Function: $p'(x) = \cos(x)$

Function to Differentiate: $r(x) = \tan(x)$

Derivative of the Function: $r'(x) = \sec^2(x)$

Function to Differentiate: $v(x) = \cosh(x)$

Derivative of the Function: $v'(x) = \sinh(x)$

Function to Differentiate: $h(x) = \ln(x)$

Derivative of the Function: $h'(x) = \frac{1}{x}$

Function to Differentiate: $m(x) = \frac{1}{\cos(x)}$

Derivative of the Function: $m'(x) = \frac{\sin(x)}{\cos^2(x)}$

Function to Differentiate: $n(x) = \arctan(x)$

Derivative of the Function: $n'(x) = \frac{1}{1 + x^2}$

Function to Differentiate: $g(x) = e^x$

Derivative of the Function: $g'(x) = e^x$

Function to Differentiate: $y(x) = \cos^2(x)$

Derivative of the Function: $y'(x) = -2\sin(x)\cos(x)$

Function to Differentiate: $j(x) = \arcsin(x)$

Derivative of the Function: $j'(x) = \frac{1}{\sqrt{1 - x^2}}$

Function to Differentiate: $u(x) = \sinh(x)$

Derivative of the Function: $u'(x) = \cosh(x)$

Function to Differentiate: $b(x) = \cosh^{-1}(x)$

Derivative of the Function: $b'(x) = \frac{1}{\sqrt{x^2 - 1}}$

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