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

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

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

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Graph of f(x) = sin(x) * cos(x) and its derivative-3-2-1123-1-0.50.511.52slope -0.4161f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) * cos(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -0.41615.

How to work it out

  1. 1

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

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

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=sin⁡(x)u = \sin\left(x\right) and v=cos⁡(x)v = \cos\left(x\right).

    u=sin⁡(x),v=cos⁡(x)u = \sin\left(x\right),\quad v = \cos\left(x\right)
  3. 3

    Differentiate each piece:

    u′=cos⁡(x),v′=−sin⁡(x)u' = \cos\left(x\right),\quad v' = -\sin\left(x\right)
  4. 4

    Put them in the rule:

    f′(x)=(cos⁡(x))(cos⁡(x))+(sin⁡(x))(−sin⁡(x))f'(x) = \left(\cos\left(x\right)\right)\left(\cos\left(x\right)\right) + \left(\sin\left(x\right)\right)\left(-\sin\left(x\right)\right)
  5. 5

    Simplify:

    f′(x)=cos⁡(x)cos⁡(x)+sin⁡(x)−sin⁡(x)f'(x) = \cos\left(x\right) \cos\left(x\right) + \sin\left(x\right) -\sin\left(x\right)
  6. 6

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

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

Values of f and f' for sin(x) * cos(x)

xf(x)f'(x)
-20.37840125-0.65364362
-1-0.45464871-0.41614684
0.50.420735490.54030231
10.45464871-0.41614684
2-0.37840125-0.65364362
3-0.139707750.96017029

Questions

What is the derivative of sin(x) * cos(x)?

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

What is the second derivative of sin(x) * cos(x)?

-(sin(x) * cos(x)) + cos(x) * -sin(x) + cos(x) * -sin(x) + sin(x) * -cos(x).

Which rule do you use for sin(x) * cos(x)?

The product rule.

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

-0.41614684.

Related derivatives

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