Derivative of cos(x)
Answer
Derivative of cos(x)
- Rule Used
- Trigonometric Rule
- Steps
- Recognize the trigonometric function . These six derivatives are standard; a cofunction (cos, cot, csc) picks up a minus sign.
- Apply the standard derivative: = -.
- Final answer: -
d/dx 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)
Derivative of cos(x) = -sin(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-cos-x
Derivative
cos(x) → -sin(x)engine: -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
Cosine is sine shifted, and the shift puts a descending slope at the origin. The derivative is −sin(x). The minus is not optional decoration: cos(x) is already decreasing at x = 0, where sin(0) = 0 but the slope of cosine is 0 only at the peaks, and just to the right of zero cosine is falling.
Apply the trigonometric derivative to cos(x). The form to remember, already simplified, is -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 -sin(x). That number is the slope of cos(x) there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = cos(x).
- Apply the trigonometric derivative to cos(x). The form to remember, already simplified, is -sin(x).
- At x = 0, substitute into the simplified form -sin(x). That number is the slope of cos(x) there; a secant over a small step in x lands next to it.
Common mistakes
Answering sin(x) instead of −sin(x)
sin(x) without the minus is the derivative of sine, read in the wrong direction. The cofunction is the one that carries the minus.
Swapping cos(x) with its cofunction
Sine and cosine, tangent and cotangent, secant and cosecant are not interchangeable. The minus sign lives on the cofunction side.
Related rule and examples
- 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(x) — Same family of differentiation, different function.
- Derivative of tan(x) — Same family of differentiation, different function.
- Derivative of sec(x) — Same family of differentiation, different function.
- Derivative of csc(x) — Same family of differentiation, different function.
- Derivative of cot(x) — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of cos(x)?
-sin(x). Cosine is sine shifted, and the shift puts a descending slope at the origin. The derivative is −sin(x). The minus is not optional decoration: cos(x) is already decreasing at x = 0, where sin(0) = 0 but the slope of cosine is 0 only at the peaks, and just to the right of zero cosine is falling.
Which wrong answer shows up for cos(x)?
sin(x) without the minus is the derivative of sine, read in the wrong direction. The cofunction is the one that carries the minus.
Where is the derivative of cos(x) defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.