## Derivative Rules

 Rule Function Derivative Basic Constant Function Rule $$f(x) = k$$ $$f'(x) = 0$$ Linear Function Rule $$f(x) = x$$ $$f'(x) = 1$$ Sum / Difference $$h = f \pm g$$ $$h' = f' \pm g'$$ $$[c + f(x)]' = f(x)'$$ Product Rule $$h = fg$$ $$h' = f'g + fg'$$ $$[kg(x)]' = kg'(x)$$ Extended Product Rule $$p = fgh$$ $$p' = f'gh + fg'h + fgh'$$ Power Rule $$f = g^n$$ $$f = ng^{~n-1}g'$$ $$x^n \Rightarrow nx^{n-1}$$ Quotient Rule $$h = {f \over g}$$ $$h' = {f'g-fg' \over g^2}$$$$,~g \ne 0$$ Chain Rule $$h(x) = f(g(x))$$ $$h'(x) = f'(g(x)) \times g'(x)$$ Exponent Function $$f(x) = b^{~g(x)}$$ $$f'(x) = b^{~g(x)} \ln b \times g'(x)$$ $$b^{~x} \Rightarrow b^{~x} \ln b$$ $$e$$ $$f(x) = e^x$$ $$f'(x) = e^x$$$$e = \lim \limits_{n \to \infty } {(1+ {1 \over n})^n}~,~\ln x = \log_{e}x$$ Logarithm $$f(x) = \log_{b} g(x)$$ $$f'(x) = {g'(x) \over g(x) \ln b}$$ (chain rule) $$\log_{b} x \Rightarrow {1 \over x \ln b}$$ Natural Logarithm $$f(x) = \ln g(x)$$ $$f'(x) = {g'(x) \over g(x)}$$ $$\ln x \Rightarrow {1 \over x}$$ Sine Function $$f(x) = \sin g(x)$$ $$f'(x) = \cos g(x) \times g'(x)$$ $$\sin x \Rightarrow \cos x$$ Cosine Function $$f(x) = \cos g(x)$$ $$f'(x) = \text{-} \sin g(x) \times g'(x)$$ $$\cos x \Rightarrow \text{-} \sin x$$ Tangent Function $$f(x) = \tan g(x)$$ $$f'(x) = \sec^2 g(x) \times g'(x)$$ $$\tan x \Rightarrow \sec^2 x$$ Cosecant Function $$f(x) = \csc x$$ $$f'(x) = \text{-} \csc x \cot x$$use radians Secant Function $$f(x) = \sec x$$ $$f'(x) = \sec x \tan x$$use radians Cotangent Function $$f(x) = \cot x$$ $$f'(x) = \text{-} \csc^2 x$$use radians arcSine Function $$f(x) = \arcsin x$$ $$f'(x) = \frac{1}{\sqrt{1-x^2}}$$ arcCosine Function $$f(x) = \arccos x$$ $$f'(x) = \text{-} \frac{1}{\sqrt{1-x^2}}$$ arcTangent Function $$f(x) = \arctan x$$ $$f'(x) = \frac{1}{x^2+1}$$

$$m_{\text{tangent}} = \text{IROC} = \lim \limits_{h \to 0} {f(a+h)-f(a) \over h} = f'(a)$$

$$m_{\text{secant}} = \text{AROC} = {f(x_{2})-f(x_{1}) \over x_{2} - x_{1} }$$