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Derivative of x2cos⁡(x)x^{2} \cos\left(x\right)

The derivative of x2cos⁡(x)x^{2} \cos\left(x\right) is 2xcos⁡(x)+x2−sin⁡(x)2 x \cos\left(x\right) + x^{2} -\sin\left(x\right).

f(x) =
ddx[x2cos⁡(x)]=2xcos⁡(x)+x2−sin⁡(x)\frac{d}{dx}\left[x^{2} \cos\left(x\right)\right] = 2 x \cos\left(x\right) + x^{2} -\sin\left(x\right)

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Graph of f(x) = x^2 * cos(x) and its derivative-3-2-1123-10-8-6-4-22468slope 0.2391f(x)f'(x)tangent at x = 1
Graph of f(x) = x^2 * cos(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.23913.

How to work it out

  1. 1

    We differentiate f(x)=x2cos⁡(x)f(x) = x^{2} \cos\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x2cos⁡(x)f(x) = x^{2} \cos\left(x\right)
  2. 2

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

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    f′(x)=2xcos⁡(x)+x2−sin⁡(x)f'(x) = 2 x \cos\left(x\right) + x^{2} -\sin\left(x\right)
  6. 6

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

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

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

xf(x)f'(x)
-2-1.66458735.3017771
-10.54030231-0.23913363
0.50.219395640.75772618
10.540302310.23913363
2-1.6645873-5.3017771
3-8.9099325-7.2100351

Questions

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

The derivative of x2cos⁡(x)x^{2} \cos\left(x\right) is 2xcos⁡(x)+x2−sin⁡(x)2 x \cos\left(x\right) + x^{2} -\sin\left(x\right).

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

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

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

The product rule.

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

0.23913363.

Related derivatives

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