Derivative of `cos(x^2)`

Step-by-step differentiation

Answer

\frac{d}{d x} [cos (x^{2})]

- (2 x \cdot sin (x^{2}))

Step-by-step solution

1Chain Rule — cos
$\frac{d}{d x} [cos (x^{2})] = - sin (x^{2}) \cdot \frac{d}{d x} [x^{2}]$
This is $cos$ applied to $u = x^{2}$ . The outer derivative is $- sin (u)$ , which we multiply by $u^{'}$ (chain rule).
↳Find u′: Differentiate the inner function, $u = x^{2}$ :
2Power Rule
$\frac{d}{d x} [x^{2}] = 2 \cdot x^{2 - 1}$
The exponent is $2$ . Bring it down in front as a coefficient, then reduce the power by $1$ .
=Combine
$\frac{d}{d x} [cos (x^{2})] = - sin (x^{2}) \cdot 2 x = - (2 sin (x^{2}) \cdot x)$
With $u^{'} = 2 x$ , multiply by the outer derivative to complete the chain rule.
✓Final Answer
$\frac{d}{d x} [cos (x^{2})] = - (2 x \cdot sin (x^{2}))$
The first derivative simplifies to $- (2 x \cdot sin (x^{2}))$ .

Understanding this derivative

Differentiating cos(x²) exercises 3 distinct techniques — the chain rule, the power rule and the trigonometric derivatives. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

The chain rule handles the composite structure here: differentiate the outer function while leaving the inner expression alone, then multiply by the derivative of that inner expression. Forgetting that final multiplication is one of the most common mistakes in differentiation. The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. The trigonometric derivatives follow the standard cycle: sine differentiates to cosine, cosine to negative sine, and tangent to secant squared. The sign on the cosine derivative is a frequent source of errors, so it is worth double-checking.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

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Derivative of cos(x^2)

Step-by-step solution

Understanding this derivative

Derivative of `cos(x^2)`