9.6PureStretch

Differentiating Trigonometric Functions

Combining the quotient and chain rules gives the derivatives of tan, sec, cosec and cot. These standard results should be memorised for exam speed.

26 min Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.6 Differentiating Trigonometric Functions (Sec, Cosec, Cot and Tan)Watch the full walkthrough before the notes below.

What you'll be able to do

The standard derivatives

These follow from the quotient rule applied to the reciprocal functions:

\frac{d}{d x} (tan x) = sec^{2} x

Tangent.

\frac{d}{d x} (sec x) = sec x tan x

Secant.

\frac{d}{d x} (cosec x) = - cosec x cot x

Cosecant.

\frac{d}{d x} (cot x) = - cosec^{2} x

Cotangent.

For a function inside, multiply by its derivative, e.g. $\frac{d}{d x} tan (3 x) = 3 sec^{2} (3 x)$ .

1Outer:

sec u \to sec u tan u

. Inner:

u = 2 x

u^{'} = 2

\frac{d y}{d x} = 2 sec 2 x tan 2 x

Answer

2 sec 2 x tan 2 x

Tip — The “co” functions (cosec, cot) carry a minus sign.

Formula recap

\frac{d}{d x} (tan x) = sec^{2} x

tan.

\frac{d}{d x} (sec x) = sec x tan x

sec.

\frac{d}{d x} (cot x) = - cosec^{2} x

cot (minus).

Common mistakes to avoid

Giving d/dx(cot x) = cosec²x (no minus).

It is −cosec²x.

Forgetting the chain rule on tan(kx) etc.

Multiply by the derivative of the inside.

Key takeaways

Test yourself

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