WordGPT

Derivative of x3x2\frac{x^{3}}{x^{2}}

The derivative of x3x2\frac{x^{3}}{x^{2}} is 3x2x2−x32x(x2)2\frac{3 x^{2} x^{2} - x^{3} 2 x}{\left(x^{2}\right)^{2}}.

f(x) =
ddx[x3x2]=3x2x2−x32x(x2)2\frac{d}{dx}\left[\frac{x^{3}}{x^{2}}\right] = \frac{3 x^{2} x^{2} - x^{3} 2 x}{\left(x^{2}\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 / x^2 and its derivative-3-2-1123-3-2-1123slope 1f(x)f'(x)tangent at x = 1
Graph of f(x) = x^3 / x^2 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.

How to work it out

  1. 1

    We differentiate f(x)=x3x2f(x) = \frac{x^{3}}{x^{2}} with respect to x. The rule that applies is the quotient rule.

    f(x)=x3x2f(x) = \frac{x^{3}}{x^{2}}
  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=x2v = x^{2}.

    u=x3,v=x2u = x^{3},\quad v = x^{2}
  3. 3

    Differentiate each piece:

    u′=3x2,v′=2xu' = 3 x^{2},\quad v' = 2 x
  4. 4

    Put them in the rule:

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

    Simplify:

    f′(x)=3x2x2−x32x(x2)2f'(x) = \frac{3 x^{2} x^{2} - x^{3} 2 x}{\left(x^{2}\right)^{2}}
  6. 6

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

    f′′(x)=(6xx2+3x22x−(3x22x+2x3))(x2)2−(3x2x2−x32x)2x22x((x2)2)2f''(x) = \frac{\left(6 x x^{2} + 3 x^{2} 2 x - \left(3 x^{2} 2 x + 2 x^{3}\right)\right) \left(x^{2}\right)^{2} - \left(3 x^{2} x^{2} - x^{3} 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{3}}{x^{2}}\right] = \frac{3 x^{2} x^{2} - x^{3} 2 x}{\left(x^{2}\right)^{2}} \]
\[ f(x) = \frac{x^{3}}{x^{2}} \]
\[ u = x^{3},\quad v = x^{2} \]
\[ u' = 3 x^{2},\quad v' = 2 x \]
\[ f'(x) = \frac{\left(3 x^{2}\right)\left(x^{2}\right) - \left(x^{3}\right)\left(2 x\right)}{\left(x^{2}\right)^{2}} \]
\[ f'(x) = \frac{3 x^{2} x^{2} - x^{3} 2 x}{\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{\left(6 x x^{2} + 3 x^{2} 2 x - \left(3 x^{2} 2 x + 2 x^{3}\right)\right) \left(x^{2}\right)^{2} - \left(3 x^{2} x^{2} - x^{3} 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}} \]

Values of f and f' for x3x^{3} / x2x^{2}

xf(x)f'(x)
-2-21
-1-11
0.50.51
111
221
331

Questions

What is the derivative of x3x^{3} / x2x^{2}?

The derivative of x3x2\frac{x^{3}}{x^{2}} is 3x2x2−x32x(x2)2\frac{3 x^{2} x^{2} - x^{3} 2 x}{\left(x^{2}\right)^{2}}.

What is the second derivative of x3x^{3} / x2x^{2}?

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

Which rule do you use for x3x^{3} / x2x^{2}?

The quotient rule.

What is the slope of x3x^{3} / x2x^{2} at x = 1?

1.

Related derivatives

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