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

The derivative of x5sin⁡(x)\frac{x^{5}}{\sin\left(x\right)} is 5x4sin⁡(x)−x5cos⁡(x)sin⁡(x)2\frac{5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

f(x) =
ddx[x5sin⁡(x)]=5x4sin⁡(x)−x5cos⁡(x)sin⁡(x)2\frac{d}{dx}\left[\frac{x^{5}}{\sin\left(x\right)}\right] = \frac{5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)}{\sin\left(x\right)^{2}}

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Graph of f(x) = x^5 / sin(x) and its derivative-3-2-1123-600-400-20020040060080010001200slope 5.179f(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) = 5.1789.

How to work it out

  1. 1

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

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

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. 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))(sin⁡(x))2f'(x) = \frac{\left(5 x^{4}\right)\left(\sin\left(x\right)\right) - \left(x^{5}\right)\left(\cos\left(x\right)\right)}{\left(\sin\left(x\right)\right)^{2}}
  5. 5

    Simplify:

    f′(x)=5x4sin⁡(x)−x5cos⁡(x)sin⁡(x)2f'(x) = \frac{5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)}{\sin\left(x\right)^{2}}
  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)))sin⁡(x)2−(5x4sin⁡(x)−x5cos⁡(x))2sin⁡(x)cos⁡(x)(sin⁡(x)2)2f''(x) = \frac{\left(20 x^{3} \sin\left(x\right) + 5 x^{4} \cos\left(x\right) - \left(5 x^{4} \cos\left(x\right) + x^{5} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)\right) 2 \sin\left(x\right) \cos\left(x\right)}{\left(\sin\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{5}}{\sin\left(x\right)}\right] = \frac{5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f(x) = \frac{x^{5}}{\sin\left(x\right)} \]
\[ u = x^{5},\quad v = \sin\left(x\right) \]
\[ u' = 5 x^{4},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \frac{\left(5 x^{4}\right)\left(\sin\left(x\right)\right) - \left(x^{5}\right)\left(\cos\left(x\right)\right)}{\left(\sin\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(20 x^{3} \sin\left(x\right) + 5 x^{4} \cos\left(x\right) - \left(5 x^{4} \cos\left(x\right) + x^{5} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)\right) 2 \sin\left(x\right) \cos\left(x\right)}{\left(\sin\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-235.192005-104.0859
-11.1883951-5.1789158
0.50.0651821760.53250659
11.18839515.1789158
235.192005104.0859
31721.938714949.732

Questions

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

The derivative of x5sin⁡(x)\frac{x^{5}}{\sin\left(x\right)} is 5x4sin⁡(x)−x5cos⁡(x)sin⁡(x)2\frac{5 x^{4} \sin\left(x\right) - x^{5} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

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))) * sin(x)^2 - (5 * x4x^{4} * sin(x) - x5x^{5} * cos(x)) * 2sin(x) * cos(x)) / (sin(x)^2)^2.

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

The quotient rule.

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

5.1789158.

Related derivatives

    Derivative of x^5 / sin(x): (5 * x^4 * sin(x) - x^5 * cos(x)) / sin(x)^2 (with steps) | WordGPT