Differentiating sin x and cos x

Video lesson · from 29:022 worked examples

One lesson video covers all of Differentiation, so it opens at 29:02 for this topic — not from the beginning.

The derivatives of the core trigonometric functions are:

\text{If } f(x) = \sin x \text{ then } f'(x) = \cos x

\text{If } f(x) = \cos x \text{ then } f'(x) = -\sin x

Wait, if the angle is in degrees, we must convert to radians to differentiate. Always assume $x$ is in radians for calculus.

The minus sign belongs to cosine: $\sin x$ differentiates to $\cos x$ , but $\cos x$ differentiates to $-\sin x$ . Losing that minus is the single most common error here.
These rules need radians: they are only valid when $x$ is in radians, so work in radians throughout a calculus question.
Keep coefficients: $4\sin x$ differentiates to $4\cos x$ — the multiplier stays.
Do not confuse with integration: the minus sign moves to the other function when integrating. Check which operation you are doing.

Example 1

Differentiate $y = 3\sin x$ with respect to $x$ .

\frac{dy}{dx} = 3\cos x

Example 2

A function $f$ is defined by $f(x) = \sin x - 2\cos x$ for $x \in \mathbb{R}$ . Find $f'(x)$ .

\begin{aligned} f(x) &= \sin x - 2\cos x \\ f'(x) &= \cos x - 2(-\sin x) \\ &= \cos x + 2\sin x \end{aligned}