Derivative of cos(x)^2

Answer

-2 \cdot \mathrm{cos}\left(x\right) \cdot \mathrm{sin}\left(x\right)

Derivative of cos(x)^2

Rule Used: Chain Rule
Steps: Identify a composite function in $\cos(x)^2$ : use the chain rule $\frac{d}{dx} f(g(x))$ = f'(g(x))·g'(x). The inner derivative is required because a stretch or shift of the input changes how fast the output moves.
Differentiate the outer function, then multiply by the inner derivative.
Final answer: - $2 \cdot c$ os(x)* $\sin(x)$

d/dx cos^2(x) = -2*cos(x)*sin(x). The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

Instant step-by-step solution for cos(x)^2

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Derivative of cos(x)^2 = -2*cos(x)*sin(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-cos-squared-x

Derivative

cos^2(x) → -2*cos(x)*sin(x)engine: -2*cos(x)*sin(x)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

cos^2(x) is the same outer power as sin^2(x), but the inside derivative is −sin(x). The minus travels to the front and the answer is −2 cos(x) sin(x), which is −sin(2x). It is the negative of the derivative of sin^2(x). That sign is predicted by the identity sin^2 + cos^2 = 1, whose derivative must be zero, so the two derivatives are opposites.

Apply the chain rule to cos^2(x). The form to remember, already simplified, is -2*cos(x)*sin(x).

The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

At x = 0, substitute into the simplified form -2*cos(x)*sin(x). That number is the slope of cos^2(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = cos^2(x).
Apply the chain rule to cos^2(x). The form to remember, already simplified, is -2*cos(x)*sin(x).
At x = 0, substitute into the simplified form -2*cos(x)*sin(x). That number is the slope of cos^2(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering 2*cos(x)

2 cos(x) misses both the minus from cosine and the inner sine. The identity check fails immediately: this would not cancel the derivative of sin^2.

Dropping the inside derivative of cos^2(x)

If the inside is not exactly x, its derivative multiplies the outside derivative. Copying a basic trig formula onto a composed inside misses that factor.

Chain rule — The rule page lists this example and the others in the same family.
Trigonometric derivatives — The rule page lists this example and the others in the same family.
Derivatives of trigonometric functions — Same family of differentiation, different function.
Derivative of sin(2x) — Same family of differentiation, different function.
Derivative of cos(2x) — Same family of differentiation, different function.
Derivative of sin^2(x) — Same family of differentiation, different function.
Derivative of tan^2(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of cos^2(x)?

-2*cos(x)*sin(x). cos^2(x) is the same outer power as sin^2(x), but the inside derivative is −sin(x). The minus travels to the front and the answer is −2 cos(x) sin(x), which is −sin(2x). It is the negative of the derivative of sin^2(x). That sign is predicted by the identity sin^2 + cos^2 = 1, whose derivative must be zero, so the two derivatives are opposites.