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Derivative of x3sin⁡(x)x^{3} \sin\left(x\right)

The derivative of x3sin⁡(x)x^{3} \sin\left(x\right) is 3x2sin⁡(x)+x3cos⁡(x)3 x^{2} \sin\left(x\right) + x^{3} \cos\left(x\right).

f(x) =
ddx[x3sin⁡(x)]=3x2sin⁡(x)+x3cos⁡(x)\frac{d}{dx}\left[x^{3} \sin\left(x\right)\right] = 3 x^{2} \sin\left(x\right) + x^{3} \cos\left(x\right)

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Graph of f(x) = x^3 * sin(x) and its derivative-3-2-1123-10-5510slope 3.065f(x)f'(x)tangent at x = 1
Graph of f(x) = x^3 * sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.0647.

How to work it out

  1. 1

    We differentiate f(x)=x3sin⁡(x)f(x) = x^{3} \sin\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x3sin⁡(x)f(x) = x^{3} \sin\left(x\right)
  2. 2

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

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=3x2sin⁡(x)+x3cos⁡(x)f'(x) = 3 x^{2} \sin\left(x\right) + x^{3} \cos\left(x\right)
  6. 6

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

    f′′(x)=6xsin⁡(x)+3x2cos⁡(x)+3x2cos⁡(x)+x3−sin⁡(x)f''(x) = 6 x \sin\left(x\right) + 3 x^{2} \cos\left(x\right) + 3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{3} \sin\left(x\right)\right] = 3 x^{2} \sin\left(x\right) + x^{3} \cos\left(x\right) \]
\[ f(x) = x^{3} \sin\left(x\right) \]
\[ u = x^{3},\quad v = \sin\left(x\right) \]
\[ u' = 3 x^{2},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \left(3 x^{2}\right)\left(\sin\left(x\right)\right) + \left(x^{3}\right)\left(\cos\left(x\right)\right) \]
\[ f'(x) = 3 x^{2} \sin\left(x\right) + x^{3} \cos\left(x\right) \]
\[ f''(x) = 6 x \sin\left(x\right) + 3 x^{2} \cos\left(x\right) + 3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right) \]

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

xf(x)f'(x)
-27.2743794-7.5823944
-10.84147098-3.0647153
0.50.0599281920.46926697
10.841470983.0647153
27.27437947.5823944
33.8102402-22.919557

Questions

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

The derivative of x3sin⁡(x)x^{3} \sin\left(x\right) is 3x2sin⁡(x)+x3cos⁡(x)3 x^{2} \sin\left(x\right) + x^{3} \cos\left(x\right).

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

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

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

The product rule.

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

3.0647153.

Related derivatives

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