Establish and use the derivatives of sin x, cos x and tan x, including with the chain, product and quotient rules.

What this dot point is asking

This dot point completes your library of standard derivatives by adding the three core trigonometric functions. Once you have these alongside powers, exponentials and logarithms, you can differentiate almost any function in the course.

The three standard derivatives

The results you must memorise are short, but the minus sign and the radian requirement trip up many students.

Notice the symmetry: differentiating sine gives cosine, and differentiating cosine gives negative sine. Differentiating twice more returns you to where you started, so the fourth derivative of $\sin x$ is $\sin x$ again.

Combining with the chain rule

For a function of the form $\sin(f(x))$, the chain rule multiplies by the derivative of the inside:

So $\dfrac{d}{dx}\sin(3x) = 3\cos(3x)$ and $\dfrac{d}{dx}\cos(x^2) = -2x\sin(x^2)$. The inner derivative is exactly the factor students most often drop.

Combining with product and quotient rules

Trigonometric terms appear inside products and quotients constantly. Apply the product rule $u'v + uv'$ or the quotient rule $\dfrac{u'v - uv'}{v^2}$ exactly as for any other functions, treating $\sin x$, $\cos x$ or $\tan x$ as one of the factors.

Where these are used

The derivatives of trigonometric functions feed straight into curve sketching (finding stationary points of $y = \sin x + \cos x$, for example), into optimisation problems with periodic models, and into rates of change for anything oscillating such as a tide height or an alternating current. They also pair with integration in Unit 4, where reversing each derivative gives the corresponding antiderivative.

Summary

Learn the three standard derivatives, keep your calculator in radian mode, and treat the trigonometric function as just another piece when you apply the chain, product or quotient rule. The two recurring slips are the missing minus sign on the derivative of cosine and the missing inner factor from the chain rule, so build a habit of checking both every time.