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Derivative of x2sin⁡(x)x^{2} \sin\left(x\right)

The derivative of x2sin⁡(x)x^{2} \sin\left(x\right) is 2xsin⁡(x)+x2cos⁡(x)2 x \sin\left(x\right) + x^{2} \cos\left(x\right).

f(x) =
ddx[x2sin⁡(x)]=2xsin⁡(x)+x2cos⁡(x)\frac{d}{dx}\left[x^{2} \sin\left(x\right)\right] = 2 x \sin\left(x\right) + x^{2} \cos\left(x\right)

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Graph of f(x) = x^2 * sin(x) and its derivative-3-2-1123-8-6-4-224slope 2.223f(x)f'(x)tangent at x = 1
Graph of f(x) = x^2 * sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.2232.

How to work it out

  1. 1

    We differentiate f(x)=x2sin⁡(x)f(x) = x^{2} \sin\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x2sin⁡(x)f(x) = x^{2} \sin\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=x2u = x^{2} and v=sin⁡(x)v = \sin\left(x\right).

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

    Differentiate each piece:

    u′=2x,v′=cos⁡(x)u' = 2 x,\quad v' = \cos\left(x\right)
  4. 4

    Put them in the rule:

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

    Simplify:

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

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

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

Values of f and f' for x2x^{2} * sin(x)

xf(x)f'(x)
-2-3.63718971.9726024
-1-0.841470982.2232443
0.50.119856380.69882118
10.841470982.2232443
23.63718971.9726024
31.2700801-8.0632124

Questions

What is the derivative of x2x^{2} * sin(x)?

The derivative of x2sin⁡(x)x^{2} \sin\left(x\right) is 2xsin⁡(x)+x2cos⁡(x)2 x \sin\left(x\right) + x^{2} \cos\left(x\right).

What is the second derivative of x2x^{2} * sin(x)?

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

Which rule do you use for x2x^{2} * sin(x)?

The product rule.

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

2.2232443.

Related derivatives

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