Derivatives of the six trigonometric functions with examples

The six trig functions have a short list of derivative rules. Let’s put them in one place, then use them in examples that also need the power rule and product rule.

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Use the six trig derivative rules

These rules use radians and apply wherever the original function is defined. In $\sin \left(x\right)$, the $x$ is the input to sine. A coefficient in front, like the $4$ in $4\cos \left(x\right)$, stays as a multiplier when we differentiate.

Trigonometric functions and their derivatives

Function	Derivative
$\sin \left(x\right)$	$\cos \left(x\right)$
$\cos \left(x\right)$	$-\sin \left(x\right)$
$\tan \left(x\right)$	${\sec }^2\left(x\right)$
$\cot \left(x\right)$	$-{\csc }^2\left(x\right)$
$\sec \left(x\right)$	$\sec \left(x\right)\tan \left(x\right)$
$\csc \left(x\right)$	$-\csc \left(x\right)\cot \left(x\right)$

Keep the constant and differentiate term by term

For $y=4\cos \left(x\right)$, the derivative is $4$ times the derivative of cosine, or $-4\sin \left(x\right)$. For $y=8x^5-9\cot \left(x\right)$, use the power rule on the first term and the cotangent rule on the second.

$\begin{array}{l}{y}^{\prime }=8\left(5\right)x^4-9(-{\csc }^2(x\left)\right)\\ y^{\prime }=40x^4+9{\csc }^2\left(x\right)\end{array}$

Use the product rule when a term has two factors

Let $y=\sec \left(x\right)-7x^5\sin \left(x\right)+3\csc \left(x\right)\cos \left(x\right)$. The derivative of $\sec \left(x\right)$ is $\sec \left(x\right)\tan \left(x\right)$. For $-7x^5\sin \left(x\right)$, differentiate each factor once, keeping the other factor in place.

$(-35x^4)\sin \left(x\right)+(-7x^5)\cos \left(x\right)$

For $3\csc \left(x\right)\cos \left(x\right)$, the product rule gives $-3\csc \left(x\right)\cot \left(x\right)\cos \left(x\right)-3\csc \left(x\right)\sin \left(x\right)$. Since $\csc \left(x\right)\cos \left(x\right)=\cot \left(x\right)$ and $\csc \left(x\right)\sin \left(x\right)=1$ on this domain, it simplifies to $-3{\cot }^2\left(x\right)-3$. Now combine the three derivatives.

$\begin{array}{l}{y}^{\prime }=\sec \left(x\right)\tan \left(x\right)-35x^4\sin \left(x\right)\\ -7x^5\cos \left(x\right)-3{\cot }^2\left(x\right)-3\end{array}$

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