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Derivative of x3cos⁡(x)x^{3} \cos\left(x\right)

The derivative of x3cos⁡(x)x^{3} \cos\left(x\right) is 3x2cos⁡(x)+x3−sin⁡(x)3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right).

f(x) =
ddx[x3cos⁡(x)]=3x2cos⁡(x)+x3−sin⁡(x)\frac{d}{dx}\left[x^{3} \cos\left(x\right)\right] = 3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right)

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Graph of f(x) = x^3 * cos(x) and its derivative-3-2-1123-30-20-1010slope 0.7794f(x)f'(x)tangent at x = 1
Graph of f(x) = x^3 * cos(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.77944.

How to work it out

  1. 1

    We differentiate f(x)=x3cos⁡(x)f(x) = x^{3} \cos\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x3cos⁡(x)f(x) = x^{3} \cos\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=x3u = x^{3} and v=cos⁡(x)v = \cos\left(x\right).

    u=x3,v=cos⁡(x)u = x^{3},\quad v = \cos\left(x\right)
  3. 3

    Differentiate each piece:

    u′=3x2,v′=−sin⁡(x)u' = 3 x^{2},\quad v' = -\sin\left(x\right)
  4. 4

    Put them in the rule:

    f′(x)=(3x2)(cos⁡(x))+(x3)(−sin⁡(x))f'(x) = \left(3 x^{2}\right)\left(\cos\left(x\right)\right) + \left(x^{3}\right)\left(-\sin\left(x\right)\right)
  5. 5

    Simplify:

    f′(x)=3x2cos⁡(x)+x3−sin⁡(x)f'(x) = 3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right)
  6. 6

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

    f′′(x)=6xcos⁡(x)+3x2−sin⁡(x)+3x2−sin⁡(x)+x3−cos⁡(x)f''(x) = 6 x \cos\left(x\right) + 3 x^{2} -\sin\left(x\right) + 3 x^{2} -\sin\left(x\right) + x^{3} -\cos\left(x\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{3} \cos\left(x\right)\right] = 3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right) \]
\[ f(x) = x^{3} \cos\left(x\right) \]
\[ u = x^{3},\quad v = \cos\left(x\right) \]
\[ u' = 3 x^{2},\quad v' = -\sin\left(x\right) \]
\[ f'(x) = \left(3 x^{2}\right)\left(\cos\left(x\right)\right) + \left(x^{3}\right)\left(-\sin\left(x\right)\right) \]
\[ f'(x) = 3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right) \]
\[ f''(x) = 6 x \cos\left(x\right) + 3 x^{2} -\sin\left(x\right) + 3 x^{2} -\sin\left(x\right) + x^{3} -\cos\left(x\right) \]

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

xf(x)f'(x)
-23.3291747-12.268141
-1-0.540302310.77943593
0.50.109697820.59825873
10.540302310.77943593
2-3.3291747-12.268141
3-26.729797-30.540038

Questions

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

The derivative of x3cos⁡(x)x^{3} \cos\left(x\right) is 3x2cos⁡(x)+x3−sin⁡(x)3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right).

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

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

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

The product rule.

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

0.77943593.

Related derivatives

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