WordGPT

Derivative of x2arctan⁡(x)\frac{x^{2}}{\arctan\left(x\right)}

The derivative of x2arctan⁡(x)\frac{x^{2}}{\arctan\left(x\right)} is 2xarctan⁡(x)−x211+x2arctan⁡(x)2\frac{2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

f(x) =
ddx[x2arctan⁡(x)]=2xarctan⁡(x)−x211+x2arctan⁡(x)2\frac{d}{dx}\left[\frac{x^{2}}{\arctan\left(x\right)}\right] = \frac{2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}}{\arctan\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^2 / atan(x) and its derivative-3-2-1123-6-4-2246slope 1.736f(x)f'(x)tangent at x = 1
Graph of f(x) = x^2 / atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.7359.

How to work it out

  1. 1

    We differentiate f(x)=x2arctan⁡(x)f(x) = \frac{x^{2}}{\arctan\left(x\right)} with respect to x. The rule that applies is the quotient rule.

    f(x)=x2arctan⁡(x)f(x) = \frac{x^{2}}{\arctan\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=x2u = x^{2} and v=arctan⁡(x)v = \arctan\left(x\right).

    u=x2,v=arctan⁡(x)u = x^{2},\quad v = \arctan\left(x\right)
  3. 3

    Differentiate each piece:

    u′=2x,v′=11+x2u' = 2 x,\quad v' = \frac{1}{1 + x^{2}}
  4. 4

    Put them in the rule:

    f′(x)=(2x)(arctan⁡(x))−(x2)(11+x2)(arctan⁡(x))2f'(x) = \frac{\left(2 x\right)\left(\arctan\left(x\right)\right) - \left(x^{2}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}}
  5. 5

    Simplify:

    f′(x)=2xarctan⁡(x)−x211+x2arctan⁡(x)2f'(x) = \frac{2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}
  6. 6

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

    f′′(x)=(2arctan⁡(x)+2x11+x2−(2x11+x2+x2−(2x)(1+x2)2))arctan⁡(x)2−(2xarctan⁡(x)−x211+x2)2arctan⁡(x)11+x2(arctan⁡(x)2)2f''(x) = \frac{\left(2 \arctan\left(x\right) + 2 x \frac{1}{1 + x^{2}} - \left(2 x \frac{1}{1 + x^{2}} + x^{2} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{2}}{\arctan\left(x\right)}\right] = \frac{2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f(x) = \frac{x^{2}}{\arctan\left(x\right)} \]
\[ u = x^{2},\quad v = \arctan\left(x\right) \]
\[ u' = 2 x,\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \frac{\left(2 x\right)\left(\arctan\left(x\right)\right) - \left(x^{2}\right)\left(\frac{1}{1 + x^{2}}\right)}{\left(\arctan\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(2 \arctan\left(x\right) + 2 x \frac{1}{1 + x^{2}} - \left(2 x \frac{1}{1 + x^{2}} + x^{2} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}\right)\right) \arctan\left(x\right)^{2} - \left(2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}\right) 2 \arctan\left(x\right) \frac{1}{1 + x^{2}}}{\left(\arctan\left(x\right)^{2}\right)^{2}} \]

Values of f and f' for x2x^{2} / atan(x)

xf(x)f'(x)
-2-3.61288412.9602375
-1-1.27323951.7359096
0.50.539202611.2264442
11.27323951.7359096
23.61288412.9602375
37.20550064.2267866

Questions

What is the derivative of x2x^{2} / atan(x)?

The derivative of x2arctan⁡(x)\frac{x^{2}}{\arctan\left(x\right)} is 2xarctan⁡(x)−x211+x2arctan⁡(x)2\frac{2 x \arctan\left(x\right) - x^{2} \frac{1}{1 + x^{2}}}{\arctan\left(x\right)^{2}}.

What is the second derivative of x2x^{2} / atan(x)?

((2atan(x) + 2x * 1 / (1 + x2x^{2}) - (2x * 1 / (1 + x2x^{2}) + x2x^{2} * -(2x) / (1 + x2x^{2})^2)) * atan(x)^2 - (2x * atan(x) - x2x^{2} * 1 / (1 + x2x^{2})) * 2atan(x) * 1 / (1 + x2x^{2})) / (atan(x)^2)^2.

Which rule do you use for x2x^{2} / atan(x)?

The quotient rule.

What is the slope of x2x^{2} / atan(x) at x = 1?

1.7359096.

Related derivatives

    Derivative of x^2 / atan(x): (2x * atan(x) - x^2 * 1 / (1 + x^2)) / atan(x)^2 (with steps) | WordGPT