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Derivative of cos⁡(x2)\cos\left(x^{2}\right)

The derivative of cos⁡(x2)\cos\left(x^{2}\right) is −(sin⁡(x2)2x)-\left(\sin\left(x^{2}\right) 2 x\right).

f(x) =
ddx[cos⁡(x2)]=−(sin⁡(x2)2x)\frac{d}{dx}\left[\cos\left(x^{2}\right)\right] = -\left(\sin\left(x^{2}\right) 2 x\right)

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Graph of f(x) = cos(x^2) and its derivative-3-2-1123-4-2246slope -1.683f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(x^2) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.6829.

How to work it out

  1. 1

    We differentiate f(x)=cos⁡(x2)f(x) = \cos\left(x^{2}\right) with respect to x. The rule that applies is the chain rule.

    f(x)=cos⁡(x2)f(x) = \cos\left(x^{2}\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)=x2g(x) = x^{2}.

    g(x)=x2,g′(x)=2xg(x) = x^{2},\quad g'(x) = 2 x
  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)=−(sin⁡(x2)2x)f'(x) = -\left(\sin\left(x^{2}\right) 2 x\right)
  5. 5

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

    f′′(x)=−(cos⁡(x2)2x2x+2sin⁡(x2))f''(x) = -\left(\cos\left(x^{2}\right) 2 x 2 x + 2 \sin\left(x^{2}\right)\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\cos\left(x^{2}\right)\right] = -\left(\sin\left(x^{2}\right) 2 x\right) \]
\[ f(x) = \cos\left(x^{2}\right) \]
\[ g(x) = x^{2},\quad g'(x) = 2 x \]
\[ f'(x) = \left[-\sin\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = -\left(\sin\left(x^{2}\right) 2 x\right) \]
\[ f''(x) = -\left(\cos\left(x^{2}\right) 2 x 2 x + 2 \sin\left(x^{2}\right)\right) \]

Values of f and f' for cos(x2x^{2})

xf(x)f'(x)
-2-0.65364362-3.02721
-10.540302311.682942
0.50.96891242-0.24740396
10.54030231-1.682942
2-0.653643623.02721
3-0.91113026-2.4727109

Questions

What is the derivative of cos(x2x^{2})?

The derivative of cos⁡(x2)\cos\left(x^{2}\right) is −(sin⁡(x2)2x)-\left(\sin\left(x^{2}\right) 2 x\right).

What is the second derivative of cos(x2x^{2})?

-(cos(x2x^{2}) * 2x * 2x + 2sin(x2x^{2})).

Which rule do you use for cos(x2x^{2})?

The chain rule.

What is the slope of cos(x2x^{2}) at x = 1?

-1.682942.

Related derivatives

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