Trigonometric functions

When we differentiate trigonometric functions, the result is another trig function. The first few derivatives in the list below are worth memorizing. Don’t forget that chain rule & friends apply for all functions, not just polynomials. Given the first two rows below, you should be able to derive the rest using just the power rule and the chain rule.

Example

Let’s differentiate $\displaystyle 7 \cos^{2}{\left ( \sqrt{x^{2} - 3} \right )}$. Using the constant rule, we have

$\displaystyle \frac{d}{dx} 7 \cos^{2}{\left ( \sqrt{x^{2} - 3} \right )} = 7 \cdot \frac{d}{dx} \left ( \cos{\left ( \sqrt{x^{2} - 3} \right )} \right )^{2}$,

and using the power rule and chain rule, we have

$\displaystyle 7 \cdot 2 \cos{\left ( \sqrt{x^{2} - 3} \right )} \cdot \frac{d}{dx} \cos{\left ( \sqrt{x^{2} - 3} \right )}$.

Now we must keep using the chain rule:

$\displaystyle 7 \cdot 2 \cos{\left ( \sqrt{x^{2} - 3} \right )} \cdot - \sin{\left ( \sqrt{x^{2} - 3} \right )} \cdot \frac{d}{dx} \sqrt{x^{2} - 3}$.

This simplifies to

$\displaystyle - 7 \sin{\left ( 2 \sqrt{x^{2} - 3} \right )} \cdot \frac{d}{dx} \sqrt{x^{2} - 3}$,

but we aren’t done with the chain rule:

$\displaystyle - 7 \sin{\left ( 2 \sqrt{x^{2} - 3} \right )} \cdot \frac{1}{2 \sqrt{x^{2} - 3}} \cdot \frac{d}{dx} \left ( x^{2} - 3 \right )$.

There is finally nothing left to differentiate. We have

$\displaystyle - 7 \sin{\left ( 2 \sqrt{x^{2} - 3} \right )} \cdot \frac{1}{2 \sqrt{x^{2} - 3}} \cdot 2 x$,

and this simplifies to our final answer:

$\displaystyle \frac{- 14 x \sin{\left ( 2 \sqrt{x^{2} - 3} \right )}}{2 \sqrt{x^{2} - 3}}$.