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In what follows, $u$ is a function of $x$.
Basic Trigonometric Functions
- $\dfrac{d }{d x}\sin x = \cos x$
- $\dfrac{d }{d x}\cos x= -\sin x$
- $\dfrac{d }{d x}\tan x= \sec^2 x$
- $\dfrac{d }{d x}\cot x= -\csc^2 x$
- $\dfrac{d }{d x}\sec x= \sec x \tan x$
- $\dfrac{d }{d x}\csc x= -\csc x \cot x$
Composite Trigonometric Functions
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$\dfrac{d }{d x}\sin u= \cos u \dfrac{d u}{d x}$
Example
$\dfrac{d }{d x}\sin (2x^2+3) = \cos (2x^2+3) (4x)$
$=4x \cos (2x^2+3) $
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$\dfrac{d }{d x}\cos u= -\sin u \dfrac{d u}{d x}$
Example
$\dfrac{d }{d x}\cos (\sin x)= -\sin (\sin x) \cos x$
$= - \cos x \sin (\sin x)$
-
$\dfrac{d }{d x}\tan u= \sec^2 u \dfrac{d u}{d x}$
Example
$\dfrac{d }{d x}\tan ((x+2)^{10}) = \sec^2 ((x+2)^{10}) (10(x+2)^9)$
$=10(x+2)^9 \sec^2 ((x+2)^{10})$
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$\dfrac{d }{d x}\cot u= -\csc^2 u \dfrac{d u}{d x}$
Example
$\dfrac{d }{d x}\cot (\dfrac{1}{x}) = -\csc^2 (\dfrac{1}{x}) \dfrac{-1}{x^2}$
$=\dfrac{1}{x^2} \csc^2 (\dfrac{1}{x})$
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$\dfrac{d }{d x}\sec u= \sec u \tan u \dfrac{d u}{d x}$
Example
$\dfrac{d }{d x}\sec (2x^2+x) = \sec (2x^2+x) \tan (2x^2+x) (4x+1)$
$=(4x+1)\sec (2x^2+x) \tan (2x^2+x)$
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$\dfrac{d }{d x}\csc u= -\csc u \cot u \dfrac{d u}{d x}$
Example
$\dfrac{d }{d x}\csc ((x^2-2)^5) = -\csc ((x^2-2)^5) \cot ((x^2-2)^5) 5(x^2-2)^4(2x)$
$=-10 x (x^2-2)^4 \csc ((x^2-2)^5) \cot ((x^2-2)^5)$
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