Skill: Differentiate trigonometric functions

To differentiate trigonometric functions, use these basic rules, with angles measured in radians:

$$\frac{d}{dx}(\sin x)=\cos x$$ $$\frac{d}{dx}(\cos x)=-\sin x$$ $$\frac{d}{dx}(\tan x)=\sec^2 x$$

If the trigonometric function contains something other than just $x$, use the chain rule:

$$\frac{d}{dx}\bigl(\sin(u)\bigr)=\cos(u)\frac{du}{dx}$$

For a product of two functions, use the product rule:

$$\frac{d}{dx}(fg)=f'g+fg'$$

Example

Differentiate

$$y=x^2\sin(3x).$$

Step 1: Identify the product

The expression is the product of

$$f(x)=x^2 \qquad\text{and}\qquad g(x)=\sin(3x).$$

So we use the product rule:

$$y'=f'(x)g(x)+f(x)g'(x).$$

Step 2: Differentiate each factor

For the first factor,

$$f'(x)=\frac{d}{dx}(x^2)=2x.$$

For the second factor, use the chain rule. The outside function is $\sin$, and the inside function is $3x$:

$$\frac{d}{dx}\bigl(\sin(3x)\bigr) =\cos(3x)\cdot\frac{d}{dx}(3x) =3\cos(3x).$$

Thus,

$$g'(x)=3\cos(3x).$$

Step 3: Substitute into the product rule

$$y'=(2x)\sin(3x)+x^2\bigl(3\cos(3x)\bigr).$$

Therefore,

$$\boxed{y'=2x\sin(3x)+3x^2\cos(3x)}.$$

The derivative gives the slope of the tangent to the curve at any value of $x$.