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Derivative of cos⁡(x3)\cos\left(x^{3}\right)

The derivative of cos⁡(x3)\cos\left(x^{3}\right) is −(sin⁡(x3)3x2)-\left(\sin\left(x^{3}\right) 3 x^{2}\right).

f(x) =
ddx[cos⁡(x3)]=−(sin⁡(x3)3x2)\frac{d}{dx}\left[\cos\left(x^{3}\right)\right] = -\left(\sin\left(x^{3}\right) 3 x^{2}\right)

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Graph of f(x) = cos(x^3) and its derivative-3-2-1123-10-5510slope -2.524f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(x^3) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -2.5244.

How to work it out

  1. 1

    We differentiate f(x)=cos⁡(x3)f(x) = \cos\left(x^{3}\right) with respect to x. The rule that applies is the chain rule.

    f(x)=cos⁡(x3)f(x) = \cos\left(x^{3}\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)=x3g(x) = x^{3}.

    g(x)=x3,g′(x)=3x2g(x) = x^{3},\quad g'(x) = 3 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)=−(sin⁡(x3)3x2)f'(x) = -\left(\sin\left(x^{3}\right) 3 x^{2}\right)
  5. 5

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

    f′′(x)=−(cos⁡(x3)3x23x2+sin⁡(x3)6x)f''(x) = -\left(\cos\left(x^{3}\right) 3 x^{2} 3 x^{2} + \sin\left(x^{3}\right) 6 x\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\cos\left(x^{3}\right)\right] = -\left(\sin\left(x^{3}\right) 3 x^{2}\right) \]
\[ f(x) = \cos\left(x^{3}\right) \]
\[ g(x) = x^{3},\quad g'(x) = 3 x^{2} \]
\[ f'(x) = \left[-\sin\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = -\left(\sin\left(x^{3}\right) 3 x^{2}\right) \]
\[ f''(x) = -\left(\cos\left(x^{3}\right) 3 x^{2} 3 x^{2} + \sin\left(x^{3}\right) 6 x\right) \]

Values of f and f' for cos(x3x^{3})

xf(x)f'(x)
-2-0.1455000311.872299
-10.540302312.524413
0.50.99219767-0.09350605
10.54030231-2.524413
2-0.14550003-11.872299
3-0.29213881-25.82215

Questions

What is the derivative of cos(x3x^{3})?

The derivative of cos⁡(x3)\cos\left(x^{3}\right) is −(sin⁡(x3)3x2)-\left(\sin\left(x^{3}\right) 3 x^{2}\right).

What is the second derivative of cos(x3x^{3})?

-(cos(x3x^{3}) * 3 * x2x^{2} * 3 * x2x^{2} + sin(x3x^{3}) * 6x).

Which rule do you use for cos(x3x^{3})?

The chain rule.

What is the slope of cos(x3x^{3}) at x = 1?

-2.524413.

Related derivatives

    Derivative of cos(x^3): -(sin(x^3) * 3 * x^2) (with steps) | WordGPT