Derivative of sin(x)^2

Answer

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

Derivative of sin(x)^2

Rule Used: Chain Rule
Steps: Identify a composite function in $\sin(x)^2$ : 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 sin^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 sin(x)^2

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

Derivative

sin^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

sin^2(x) means (sin x)^2, a power on the outside and sine inside. The outer power gives 2 sin(x), and the derivative of the inside, cos(x), still multiplies, so the result is 2 sin(x) cos(x). That product is also sin(2x) by the double-angle identity, which is a useful check rather than a different answer.

Apply the chain rule to sin^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 sin^2(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = sin^2(x).
Apply the chain rule to sin^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 sin^2(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering 2*sin(x)

2 sin(x) differentiates the square and treats the inside as if it were x. Sine's derivative is cosine, and it has to appear.

Dropping the inside derivative of sin^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 cos^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 sin^2(x)?

2*cos(x)*sin(x). sin^2(x) means (sin x)^2, a power on the outside and sine inside. The outer power gives 2 sin(x), and the derivative of the inside, cos(x), still multiplies, so the result is 2 sin(x) cos(x). That product is also sin(2x) by the double-angle identity, which is a useful check rather than a different answer.