9.1PureStretch

Differentiating sin x and cos x

Using the small angle approximations and the limit definition, we can differentiate sin x and cos x from first principles. The results are clean and must be memorised — but only hold when x is in radians.

26 min Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.1 First Principles, Differentiating Sin(x) and Cos(x)Watch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Differentiate sin x and cos x
Understand the first-principles derivation
Apply the results to gradients
Know they require radians

The results

From first principles (using $sin θ \approx θ$ for small $θ$ ): the derivative of $sin x$ is $cos x$ , and the derivative of $cos x$ is $- sin x$ .

\frac{d}{d x} (sin x) = cos x

Derivative of sine.

\frac{d}{d x} (cos x) = - sin x

Derivative of cosine (note the minus).

Why radians?

The first-principles limit $lim_{h \to 0} \frac{s i n h}{h} = 1$ only holds when $h$ is in radians, so these derivatives are only valid in radians.

\frac{d y}{d x} = cos x

2At

x = 0

cos 0 = 1

AnswerGradient

= 1

Tip — Remember the minus: d/dx(cos x) = −sin x.

Formula recap

\frac{d}{d x} (sin x) = cos x

Sine.

\frac{d}{d x} (cos x) = - sin x

Cosine.

Common mistakes to avoid

Differentiating cos x to +sin x.

It is −sin x.

Using these derivatives in degrees.

They only hold for x in radians.

Key takeaways

d/dx(sin x) = cos x.
d/dx(cos x) = −sin x.
Valid only in radians (from the sin h / h limit).

Test yourself

Ready to lock in Differentiating sin x and cos x? Pick a mode and earn XP & Dobloons.