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Derivative of cos⁡(2x)\cos\left(2 x\right)

The derivative of cos⁡(2x)\cos\left(2 x\right) is −(2sin⁡(2x))-\left(2 \sin\left(2 x\right)\right).

f(x) =
ddx[cos⁡(2x)]=−(2sin⁡(2x))\frac{d}{dx}\left[\cos\left(2 x\right)\right] = -\left(2 \sin\left(2 x\right)\right)

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Graph of f(x) = cos(2x) and its derivative-3-2-1123-4-2246slope -1.819f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(2x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.8186.

How to work it out

  1. 1

    We differentiate f(x)=cos⁡(2x)f(x) = \cos\left(2 x\right) with respect to x. The rule that applies is the chain rule.

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

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=2xg(x) = 2 x.

    g(x)=2x,g′(x)=2g(x) = 2 x,\quad g'(x) = 2
  3. 3

    The outer function is cos(u); its derivative evaluated at g(x), times g'(x):

    f′(x)=[−sin⁡(u)]u=g(x)⋅g′(x)f'(x) = \left[-\sin\left(u\right)\right]_{u=g(x)} \cdot g'(x)
  4. 4

    Simplify:

    f′(x)=−(2sin⁡(2x))f'(x) = -\left(2 \sin\left(2 x\right)\right)
  5. 5

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

    f′′(x)=−(4cos⁡(2x))f''(x) = -\left(4 \cos\left(2 x\right)\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\cos\left(2 x\right)\right] = -\left(2 \sin\left(2 x\right)\right) \]
\[ f(x) = \cos\left(2 x\right) \]
\[ g(x) = 2 x,\quad g'(x) = 2 \]
\[ f'(x) = \left[-\sin\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = -\left(2 \sin\left(2 x\right)\right) \]
\[ f''(x) = -\left(4 \cos\left(2 x\right)\right) \]

Values of f and f' for cos(2x)

xf(x)f'(x)
-2-0.65364362-1.513605
-1-0.416146841.8185949
0.50.54030231-1.682942
1-0.41614684-1.8185949
2-0.653643621.513605
30.960170290.558831

Questions

What is the derivative of cos(2x)?

The derivative of cos⁡(2x)\cos\left(2 x\right) is −(2sin⁡(2x))-\left(2 \sin\left(2 x\right)\right).

What is the second derivative of cos(2x)?

-(4cos(2x)).

Which rule do you use for cos(2x)?

The chain rule.

What is the slope of cos(2x) at x = 1?

-1.8185949.

Related derivatives

    Derivative of cos(2x): -(2sin(2x)) (with steps) | WordGPT