Derivative of cos(2x)

Answer

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

Derivative of cos(2x)

Rule Used: Chain Rule
Steps: Identify a composite function in $\cos(2 \cdot x)$ : 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: - $2 \cdot s$ in( $2 \cdot x$ )

d/dx cos(2x) = -2*sin(2*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(2x)

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

Derivative

cos(2x) → -2*sin(2*x)engine: -2*sin(2*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(2x) stacks two minuses worth of structure: cosine already contributes a minus, and the inside 2x contributes a factor 2 that is positive. The product is −2 sin(2x). The 2 does not cancel the minus. Compared with sin(2x), the sign flips and sine replaces cosine, which is the same relationship as between cos(x) and sin(x), scaled by 2.

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

Worked derivative

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

Common mistakes

Answering -sin(2*x)

−sin(2x) remembers the minus from cosine and forgets that the inside still has derivative 2. The slope at a small positive x is about −4x, not −2x.

Dropping the inside derivative of cos(2x)

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 sin^2(x) — 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 cos(2x)?

-2*sin(2*x). cos(2x) stacks two minuses worth of structure: cosine already contributes a minus, and the inside 2x contributes a factor 2 that is positive. The product is −2 sin(2x). The 2 does not cancel the minus. Compared with sin(2x), the sign flips and sine replaces cosine, which is the same relationship as between cos(x) and sin(x), scaled by 2.