WordGPT

Derivative of x3sin⁡(x)\frac{x^{3}}{\sin\left(x\right)}

The derivative of x3sin⁡(x)\frac{x^{3}}{\sin\left(x\right)} is 3x2sin⁡(x)−x3cos⁡(x)sin⁡(x)2\frac{3 x^{2} \sin\left(x\right) - x^{3} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

f(x) =
ddx[x3sin⁡(x)]=3x2sin⁡(x)−x3cos⁡(x)sin⁡(x)2\frac{d}{dx}\left[\frac{x^{3}}{\sin\left(x\right)}\right] = \frac{3 x^{2} \sin\left(x\right) - x^{3} \cos\left(x\right)}{\sin\left(x\right)^{2}}

Type any function of x; the derivative is computed symbolically in your browser.

Explain it step by step in chatMake a short video of it
Graph of f(x) = x^3 / sin(x) and its derivative-3-2-1123-5050100slope 2.802f(x)f'(x)tangent at x = 1
Graph of f(x) = x^3 / sin(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.8021.

How to work it out

  1. 1

    We differentiate f(x)=x3sin⁡(x)f(x) = \frac{x^{3}}{\sin\left(x\right)} with respect to x. The rule that applies is the quotient rule.

    f(x)=x3sin⁡(x)f(x) = \frac{x^{3}}{\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=x3u = x^{3} and v=sin⁡(x)v = \sin\left(x\right).

    u=x3,v=sin⁡(x)u = x^{3},\quad v = \sin\left(x\right)
  3. 3

    Differentiate each piece:

    u′=3x2,v′=cos⁡(x)u' = 3 x^{2},\quad v' = \cos\left(x\right)
  4. 4

    Put them in the rule:

    f′(x)=(3x2)(sin⁡(x))−(x3)(cos⁡(x))(sin⁡(x))2f'(x) = \frac{\left(3 x^{2}\right)\left(\sin\left(x\right)\right) - \left(x^{3}\right)\left(\cos\left(x\right)\right)}{\left(\sin\left(x\right)\right)^{2}}
  5. 5

    Simplify:

    f′(x)=3x2sin⁡(x)−x3cos⁡(x)sin⁡(x)2f'(x) = \frac{3 x^{2} \sin\left(x\right) - x^{3} \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)=(6xsin⁡(x)+3x2cos⁡(x)−(3x2cos⁡(x)+x3−sin⁡(x)))sin⁡(x)2−(3x2sin⁡(x)−x3cos⁡(x))2sin⁡(x)cos⁡(x)(sin⁡(x)2)2f''(x) = \frac{\left(6 x \sin\left(x\right) + 3 x^{2} \cos\left(x\right) - \left(3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(3 x^{2} \sin\left(x\right) - x^{3} \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
\[ \frac{d}{dx}\left[\frac{x^{3}}{\sin\left(x\right)}\right] = \frac{3 x^{2} \sin\left(x\right) - x^{3} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f(x) = \frac{x^{3}}{\sin\left(x\right)} \]
\[ u = x^{3},\quad v = \sin\left(x\right) \]
\[ u' = 3 x^{2},\quad v' = \cos\left(x\right) \]
\[ f'(x) = \frac{\left(3 x^{2}\right)\left(\sin\left(x\right)\right) - \left(x^{3}\right)\left(\cos\left(x\right)\right)}{\left(\sin\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{3 x^{2} \sin\left(x\right) - x^{3} \cos\left(x\right)}{\sin\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(6 x \sin\left(x\right) + 3 x^{2} \cos\left(x\right) - \left(3 x^{2} \cos\left(x\right) + x^{3} -\sin\left(x\right)\right)\right) \sin\left(x\right)^{2} - \left(3 x^{2} \sin\left(x\right) - x^{3} \cos\left(x\right)\right) 2 \sin\left(x\right) \cos\left(x\right)}{\left(\sin\left(x\right)^{2}\right)^{2}} \]

Values of f and f' for x3x^{3} / sin(x)

xf(x)f'(x)
-28.7980014-17.223474
-11.1883951-2.8021256
0.50.260728711.0871115
11.18839512.8021256
28.798001417.223474
3191.326521533.5304

Questions

What is the derivative of x3x^{3} / sin(x)?

The derivative of x3sin⁡(x)\frac{x^{3}}{\sin\left(x\right)} is 3x2sin⁡(x)−x3cos⁡(x)sin⁡(x)2\frac{3 x^{2} \sin\left(x\right) - x^{3} \cos\left(x\right)}{\sin\left(x\right)^{2}}.

What is the second derivative of x3x^{3} / sin(x)?

((6x * sin(x) + 3 * x2x^{2} * cos(x) - (3 * x2x^{2} * cos(x) + x3x^{3} * -sin(x))) * sin(x)^2 - (3 * x2x^{2} * sin(x) - x3x^{3} * cos(x)) * 2sin(x) * cos(x)) / (sin(x)^2)^2.

Which rule do you use for x3x^{3} / sin(x)?

The quotient rule.

What is the slope of x3x^{3} / sin(x) at x = 1?

2.8021256.

Related derivatives

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