WordGPT

Derivative of tan⁡(x)x2\frac{\tan\left(x\right)}{x^{2}}

The derivative of tan⁡(x)x2\frac{\tan\left(x\right)}{x^{2}} is x2cos⁡(x)2−tan⁡(x)2x(x2)2\frac{\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 2 x}{\left(x^{2}\right)^{2}}.

f(x) =
ddx[tan⁡(x)x2]=x2cos⁡(x)2−tan⁡(x)2x(x2)2\frac{d}{dx}\left[\frac{\tan\left(x\right)}{x^{2}}\right] = \frac{\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 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) = tan(x) / x^2 and its derivative-3-2-1123-20-10102030slope 0.3107f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(x) / x^2 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.3107.

How to work it out

  1. 1

    We differentiate f(x)=tan⁡(x)x2f(x) = \frac{\tan\left(x\right)}{x^{2}} with respect to x. The rule that applies is the quotient rule.

    f(x)=tan⁡(x)x2f(x) = \frac{\tan\left(x\right)}{x^{2}}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=tan⁡(x)u = \tan\left(x\right) and v=x2v = x^{2}.

    u=tan⁡(x),v=x2u = \tan\left(x\right),\quad v = x^{2}
  3. 3

    Differentiate each piece:

    u′=1cos⁡(x)2,v′=2xu' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = 2 x
  4. 4

    Put them in the rule:

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

    Simplify:

    f′(x)=x2cos⁡(x)2−tan⁡(x)2x(x2)2f'(x) = \frac{\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 2 x}{\left(x^{2}\right)^{2}}
  6. 6

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

    f′′(x)=(2xcos⁡(x)2−x22cos⁡(x)−sin⁡(x)(cos⁡(x)2)2−(2xcos⁡(x)2+2tan⁡(x)))(x2)2−(x2cos⁡(x)2−tan⁡(x)2x)2x22x((x2)2)2f''(x) = \frac{\left(\frac{2 x \cos\left(x\right)^{2} - x^{2} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{2 x}{\cos\left(x\right)^{2}} + 2 \tan\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\tan\left(x\right)}{x^{2}}\right] = \frac{\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f(x) = \frac{\tan\left(x\right)}{x^{2}} \]
\[ u = \tan\left(x\right),\quad v = x^{2} \]
\[ u' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = 2 x \]
\[ f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(x^{2}\right) - \left(\tan\left(x\right)\right)\left(2 x\right)}{\left(x^{2}\right)^{2}} \]
\[ f'(x) = \frac{\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{2 x \cos\left(x\right)^{2} - x^{2} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{2 x}{\cos\left(x\right)^{2}} + 2 \tan\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-20.546259971.9898598
-1-1.55740770.31070337
0.52.18521-3.5470542
11.55740770.31070337
2-0.546259971.9898598
3-0.0158385050.12392784

Questions

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

The derivative of tan⁡(x)x2\frac{\tan\left(x\right)}{x^{2}} is x2cos⁡(x)2−tan⁡(x)2x(x2)2\frac{\frac{x^{2}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 2 x}{\left(x^{2}\right)^{2}}.

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

(((2x * cos(x)^2 - x2x^{2} * 2cos(x) * -sin(x)) / (cos(x)^2)^2 - (2x / cos(x)^2 + 2tan(x))) * (x2x^{2})^2 - (x2x^{2} / cos(x)^2 - tan(x) * 2x) * 2 * x2x^{2} * 2x) / ((x2x^{2})^2)^2.

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

The quotient rule.

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

0.31070337.

Related derivatives

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