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Derivative of cos⁡(x)\cos\left(x\right)

The derivative of cos⁡(x)\cos\left(x\right) is −sin⁡(x)-\sin\left(x\right).

f(x) =
ddx[cos⁡(x)]=−sin⁡(x)\frac{d}{dx}\left[\cos\left(x\right)\right] = -\sin\left(x\right)

Type any function of x; the derivative is computed symbolically in your browser.

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Graph of f(x) = cos(x) and its derivative-3-2-1123-11234slope -0.8415f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -0.84147.

How to work it out

  1. 1

    We differentiate f(x)=cos⁡(x)f(x) = \cos\left(x\right) with respect to x. The rule that applies is the standard derivative.

    f(x)=cos⁡(x)f(x) = \cos\left(x\right)
  2. 2

    This is a standard derivative from the table.

  3. 3

    Simplify:

    f′(x)=−sin⁡(x)f'(x) = -\sin\left(x\right)
  4. 4

    Differentiating once more gives the second derivative, which tells you about curvature:

    f′′(x)=−cos⁡(x)f''(x) = -\cos\left(x\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\cos\left(x\right)\right] = -\sin\left(x\right) \]
\[ f(x) = \cos\left(x\right) \]
\[ f'(x) = -\sin\left(x\right) \]
\[ f''(x) = -\cos\left(x\right) \]

Values of f and f' for cos(x)

xf(x)f'(x)
-2-0.416146840.90929743
-10.540302310.84147098
0.50.87758256-0.47942554
10.54030231-0.84147098
2-0.41614684-0.90929743
3-0.9899925-0.14112001

Questions

What is the derivative of cos(x)?

The derivative of cos⁡(x)\cos\left(x\right) is −sin⁡(x)-\sin\left(x\right).

What is the second derivative of cos(x)?

-cos(x).

Which rule do you use for cos(x)?

The standard derivative.

What is the slope of cos(x) at x = 1?

-0.84147098.

Related derivatives

    Derivative of cos(x): -sin(x) (with steps) | WordGPT