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Derivative of x5sin⁡(x)x^{5} \sin\left(x\right)

The derivative of x5sin⁡(x)x^{5} \sin\left(x\right) is 5x4sin⁡(x)+x5cos⁡(x)5 x^{4} \sin\left(x\right) + x^{5} \cos\left(x\right).

f(x) =
ddx[x5sin⁡(x)]=5x4sin⁡(x)+x5cos⁡(x)\frac{d}{dx}\left[x^{5} \sin\left(x\right)\right] = 5 x^{4} \sin\left(x\right) + x^{5} \cos\left(x\right)

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Graph of f(x) = x^5 * sin(x) and its derivative-3-2-1123-60-40-20204060slope 4.748f(x)f'(x)tangent at x = 1
Graph of f(x) = x^5 * sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 4.7477.

How to work it out

  1. 1

    We differentiate f(x)=x5sin⁡(x)f(x) = x^{5} \sin\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x5sin⁡(x)f(x) = x^{5} \sin\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=x5u = x^{5} and v=sin⁡(x)v = \sin\left(x\right).

    u=x5,v=sin⁡(x)u = x^{5},\quad v = \sin\left(x\right)
  3. 3

    Differentiate each piece:

    u′=5x4,v′=cos⁡(x)u' = 5 x^{4},\quad v' = \cos\left(x\right)
  4. 4

    Put them in the rule:

    f′(x)=(5x4)(sin⁡(x))+(x5)(cos⁡(x))f'(x) = \left(5 x^{4}\right)\left(\sin\left(x\right)\right) + \left(x^{5}\right)\left(\cos\left(x\right)\right)
  5. 5

    Simplify:

    f′(x)=5x4sin⁡(x)+x5cos⁡(x)f'(x) = 5 x^{4} \sin\left(x\right) + x^{5} \cos\left(x\right)
  6. 6

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

    f′′(x)=20x3sin⁡(x)+5x4cos⁡(x)+5x4cos⁡(x)+x5−sin⁡(x)f''(x) = 20 x^{3} \sin\left(x\right) + 5 x^{4} \cos\left(x\right) + 5 x^{4} \cos\left(x\right) + x^{5} -\sin\left(x\right)
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{5} \sin\left(x\right)\right] = 5 x^{4} \sin\left(x\right) + x^{5} \cos\left(x\right) \]
\[ f(x) = x^{5} \sin\left(x\right) \]
\[ u = x^{5},\quad v = \sin\left(x\right) \]
\[ u' = 5 x^{4},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \left(5 x^{4}\right)\left(\sin\left(x\right)\right) + \left(x^{5}\right)\left(\cos\left(x\right)\right) \]
\[ f'(x) = 5 x^{4} \sin\left(x\right) + x^{5} \cos\left(x\right) \]
\[ f''(x) = 20 x^{3} \sin\left(x\right) + 5 x^{4} \cos\left(x\right) + 5 x^{4} \cos\left(x\right) + x^{5} -\sin\left(x\right) \]

Values of f and f' for x5x^{5} * sin(x)

xf(x)f'(x)
-229.097518-59.427095
-10.84147098-4.7476572
0.50.0149820480.17724494
10.841470984.7476572
229.09751859.427095
334.292162-183.41457

Questions

What is the derivative of x5x^{5} * sin(x)?

The derivative of x5sin⁡(x)x^{5} \sin\left(x\right) is 5x4sin⁡(x)+x5cos⁡(x)5 x^{4} \sin\left(x\right) + x^{5} \cos\left(x\right).

What is the second derivative of x5x^{5} * sin(x)?

20 * x3x^{3} * sin(x) + 5 * x4x^{4} * cos(x) + 5 * x4x^{4} * cos(x) + x5x^{5} * -sin(x).

Which rule do you use for x5x^{5} * sin(x)?

The product rule.

What is the slope of x5x^{5} * sin(x) at x = 1?

4.7476572.

Related derivatives

    Derivative of x^5 * sin(x): 5 * x^4 * sin(x) + x^5 * cos(x) (with steps) | WordGPT