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

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

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

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

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Graph of f(x) = sin(x) and its derivative-3-2-1123-1-0.50.511.52slope 0.5403f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.5403.

How to work it out

  1. 1

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

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

    This is a standard derivative from the table.

  3. 3

    Simplify:

    f′(x)=cos⁡(x)f'(x) = \cos\left(x\right)
  4. 4

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

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

Values of f and f' for sin(x)

xf(x)f'(x)
-2-0.90929743-0.41614684
-1-0.841470980.54030231
0.50.479425540.87758256
10.841470980.54030231
20.90929743-0.41614684
30.14112001-0.9899925

Questions

What is the derivative of sin(x)?

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

What is the second derivative of sin(x)?

-sin(x).

Which rule do you use for sin(x)?

The standard derivative.

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

0.54030231.

Related derivatives

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