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Derivative of x4sin⁡(x)x^{4} \sin\left(x\right)

The derivative of x4sin⁡(x)x^{4} \sin\left(x\right) is 4x3sin⁡(x)+x4cos⁡(x)4 x^{3} \sin\left(x\right) + x^{4} \cos\left(x\right).

f(x) =
ddx[x4sin⁡(x)]=4x3sin⁡(x)+x4cos⁡(x)\frac{d}{dx}\left[x^{4} \sin\left(x\right)\right] = 4 x^{3} \sin\left(x\right) + x^{4} \cos\left(x\right)

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Graph of f(x) = x^4 * sin(x) and its derivative-3-2-1123-20-101020slope 3.906f(x)f'(x)tangent at x = 1
Graph of f(x) = x^4 * sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.9062.

How to work it out

  1. 1

    We differentiate f(x)=x4sin⁡(x)f(x) = x^{4} \sin\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x4sin⁡(x)f(x) = x^{4} \sin\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=x4u = x^{4} and v=sin⁡(x)v = \sin\left(x\right).

    u=x4,v=sin⁡(x)u = x^{4},\quad v = \sin\left(x\right)
  3. 3

    Differentiate each piece:

    u′=4x3,v′=cos⁡(x)u' = 4 x^{3},\quad v' = \cos\left(x\right)
  4. 4

    Put them in the rule:

    f′(x)=(4x3)(sin⁡(x))+(x4)(cos⁡(x))f'(x) = \left(4 x^{3}\right)\left(\sin\left(x\right)\right) + \left(x^{4}\right)\left(\cos\left(x\right)\right)
  5. 5

    Simplify:

    f′(x)=4x3sin⁡(x)+x4cos⁡(x)f'(x) = 4 x^{3} \sin\left(x\right) + x^{4} \cos\left(x\right)
  6. 6

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

    f′′(x)=12x2sin⁡(x)+4x3cos⁡(x)+4x3cos⁡(x)+x4−sin⁡(x)f''(x) = 12 x^{2} \sin\left(x\right) + 4 x^{3} \cos\left(x\right) + 4 x^{3} \cos\left(x\right) + x^{4} -\sin\left(x\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{4} \sin\left(x\right)\right] = 4 x^{3} \sin\left(x\right) + x^{4} \cos\left(x\right) \]
\[ f(x) = x^{4} \sin\left(x\right) \]
\[ u = x^{4},\quad v = \sin\left(x\right) \]
\[ u' = 4 x^{3},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \left(4 x^{3}\right)\left(\sin\left(x\right)\right) + \left(x^{4}\right)\left(\cos\left(x\right)\right) \]
\[ f'(x) = 4 x^{3} \sin\left(x\right) + x^{4} \cos\left(x\right) \]
\[ f''(x) = 12 x^{2} \sin\left(x\right) + 4 x^{3} \cos\left(x\right) + 4 x^{3} \cos\left(x\right) + x^{4} -\sin\left(x\right) \]

Values of f and f' for x4x^{4} * sin(x)

xf(x)f'(x)
-2-14.54875922.439168
-1-0.841470983.9061862
0.50.0299640960.29456168
10.841470983.9061862
214.54875922.439168
311.430721-64.948431

Questions

What is the derivative of x4x^{4} * sin(x)?

The derivative of x4sin⁡(x)x^{4} \sin\left(x\right) is 4x3sin⁡(x)+x4cos⁡(x)4 x^{3} \sin\left(x\right) + x^{4} \cos\left(x\right).

What is the second derivative of x4x^{4} * sin(x)?

12 * x2x^{2} * sin(x) + 4 * x3x^{3} * cos(x) + 4 * x3x^{3} * cos(x) + x4x^{4} * -sin(x).

Which rule do you use for x4x^{4} * sin(x)?

The product rule.

What is the slope of x4x^{4} * sin(x) at x = 1?

3.9061862.

Related derivatives

    Derivative of x^4 * sin(x): 4 * x^3 * sin(x) + x^4 * cos(x) (with steps) | WordGPT