Math Doubts

Derivatives of Trigonometric functions

In differential calculus, there are six derivative formulas to find the differentiation of the trigonometric functions. Each derivative rule is given here with mathematical proof.

Sin function

The derivative of $\sin{x}$ with respect $x$ is equal to $\cos{x}$.

$\dfrac{d}{dx}{\sin{x}} \,=\, \cos{x}$

Cos function

The derivative of $\cos{x}$ with respect $x$ is equal to negative $\sin{x}$.

$\dfrac{d}{dx}{\cos{x}} \,=\, -\sin{x}$

Tan function

The derivative of $\tan{x}$ with respect $x$ is equal to square of $\sec{x}$.

$\dfrac{d}{dx}{\tan{x}} \,=\, \sec^2{x}$

Cot function

The derivative of $\cot{x}$ with respect $x$ is equal to negative of square of $\csc{x}$.

$\dfrac{d}{dx}{\cot{x}} \,=\, -\csc^2{x}$ (or) $-\operatorname{cosec}^2{x}$

Sec function

The derivative of $\sec{x}$ with respect $x$ is equal to product of $\sec{x}$ and $\tan{x}$.

$\dfrac{d}{dx}{\sec{x}} \,=\, \sec{x}.\tan{x}$

Csc (or) Cosec function

The derivative of $\csc{x}$ with respect $x$ is equal to negative of product of $\csc{x}$ and $\cot{x}$.

$\dfrac{d}{dx}{\csc{x}} \,=\, -\csc{x}.\cot{x}$ (or) $-\operatorname{cosec}{x}.\cot{x}$

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