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

The derivative of x4sin⁡(x)\frac{x^{4}}{\sin\left(x\right)} is 4x3sin⁡(x)−x4cos⁡(x)sin⁡(x)2\frac{4 x^{3} \sin\left(x\right) - x^{4} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

f(x) =
ddx[x4sin⁡(x)]=4x3sin⁡(x)−x4cos⁡(x)sin⁡(x)2\frac{d}{dx}\left[\frac{x^{4}}{\sin\left(x\right)}\right] = \frac{4 x^{3} \sin\left(x\right) - x^{4} \cos\left(x\right)}{\sin\left(x\right)^{2}}

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Graph of f(x) = x^4 / sin(x) and its derivative-3-2-1123-100100200300400500600slope 3.991f(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) = 3.9905.

How to work it out

  1. 1

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

    f(x)=x4sin⁡(x)f(x) = \frac{x^{4}}{\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=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))(sin⁡(x))2f'(x) = \frac{\left(4 x^{3}\right)\left(\sin\left(x\right)\right) - \left(x^{4}\right)\left(\cos\left(x\right)\right)}{\left(\sin\left(x\right)\right)^{2}}
  5. 5

    Simplify:

    f′(x)=4x3sin⁡(x)−x4cos⁡(x)sin⁡(x)2f'(x) = \frac{4 x^{3} \sin\left(x\right) - x^{4} \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)=(12x2sin⁡(x)+4x3cos⁡(x)−(4x3cos⁡(x)+x4−sin⁡(x)))sin⁡(x)2−(4x3sin⁡(x)−x4cos⁡(x))2sin⁡(x)cos⁡(x)(sin⁡(x)2)2f''(x) = \frac{\left(12 x^{2} \sin\left(x\right) + 4 x^{3} \cos\left(x\right) - \left(4 x^{3} \cos\left(x\right) + x^{4} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(4 x^{3} \sin\left(x\right) - x^{4} \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
\[ \frac{d}{dx}\left[\frac{x^{4}}{\sin\left(x\right)}\right] = \frac{4 x^{3} \sin\left(x\right) - x^{4} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f(x) = \frac{x^{4}}{\sin\left(x\right)} \]
\[ u = x^{4},\quad v = \sin\left(x\right) \]
\[ u' = 4 x^{3},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \frac{\left(4 x^{3}\right)\left(\sin\left(x\right)\right) - \left(x^{4}\right)\left(\cos\left(x\right)\right)}{\left(\sin\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{4 x^{3} \sin\left(x\right) - x^{4} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(12 x^{2} \sin\left(x\right) + 4 x^{3} \cos\left(x\right) - \left(4 x^{3} \cos\left(x\right) + x^{4} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(4 x^{3} \sin\left(x\right) - x^{4} \cos\left(x\right)\right) 2 \sin\left(x\right) \cos\left(x\right)}{\left(\sin\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-17.59600343.244949
-1-1.18839513.9905207
0.50.130364350.80428447
11.18839513.9905207
217.59600343.244949
3573.979564791.9176

Questions

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

The derivative of x4sin⁡(x)\frac{x^{4}}{\sin\left(x\right)} is 4x3sin⁡(x)−x4cos⁡(x)sin⁡(x)2\frac{4 x^{3} \sin\left(x\right) - x^{4} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

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

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

The quotient rule.

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

3.9905207.

Related derivatives

    Derivative of x^4 / sin(x): (4 * x^3 * sin(x) - x^4 * cos(x)) / sin(x)^2 (with steps) | WordGPT