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Derivative of tan⁡(x)x4\frac{\tan\left(x\right)}{x^{4}}

The derivative of tan⁡(x)x4\frac{\tan\left(x\right)}{x^{4}} is x4cos⁡(x)2−tan⁡(x)4x3(x4)2\frac{\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}.

f(x) =
ddx[tan⁡(x)x4]=x4cos⁡(x)2−tan⁡(x)4x3(x4)2\frac{d}{dx}\left[\frac{\tan\left(x\right)}{x^{4}}\right] = \frac{\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}

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Graph of f(x) = tan(x) / x^4 and its derivative-3-2-1123-1000-800-600-400-200slope -2.804f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(x) / x^4 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -2.8041.

How to work it out

  1. 1

    We differentiate f(x)=tan⁡(x)x4f(x) = \frac{\tan\left(x\right)}{x^{4}} with respect to x. The rule that applies is the quotient rule.

    f(x)=tan⁡(x)x4f(x) = \frac{\tan\left(x\right)}{x^{4}}
  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=x4v = x^{4}.

    u=tan⁡(x),v=x4u = \tan\left(x\right),\quad v = x^{4}
  3. 3

    Differentiate each piece:

    u′=1cos⁡(x)2,v′=4x3u' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = 4 x^{3}
  4. 4

    Put them in the rule:

    f′(x)=(1cos⁡(x)2)(x4)−(tan⁡(x))(4x3)(x4)2f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(x^{4}\right) - \left(\tan\left(x\right)\right)\left(4 x^{3}\right)}{\left(x^{4}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=x4cos⁡(x)2−tan⁡(x)4x3(x4)2f'(x) = \frac{\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}
  6. 6

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

    f′′(x)=(4x3cos⁡(x)2−x42cos⁡(x)−sin⁡(x)(cos⁡(x)2)2−(4x3cos⁡(x)2+tan⁡(x)12x2))(x4)2−(x4cos⁡(x)2−tan⁡(x)4x3)2x44x3((x4)2)2f''(x) = \frac{\left(\frac{4 x^{3} \cos\left(x\right)^{2} - x^{4} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{4 x^{3}}{\cos\left(x\right)^{2}} + \tan\left(x\right) 12 x^{2}\right)\right) \left(x^{4}\right)^{2} - \left(\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}\right) 2 x^{4} 4 x^{3}}{\left(\left(x^{4}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\tan\left(x\right)}{x^{4}}\right] = \frac{\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}} \]
\[ f(x) = \frac{\tan\left(x\right)}{x^{4}} \]
\[ u = \tan\left(x\right),\quad v = x^{4} \]
\[ u' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = 4 x^{3} \]
\[ f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(x^{4}\right) - \left(\tan\left(x\right)\right)\left(4 x^{3}\right)}{\left(x^{4}\right)^{2}} \]
\[ f'(x) = \frac{\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{4 x^{3} \cos\left(x\right)^{2} - x^{4} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{4 x^{3}}{\cos\left(x\right)^{2}} + \tan\left(x\right) 12 x^{2}\right)\right) \left(x^{4}\right)^{2} - \left(\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}\right) 2 x^{4} 4 x^{3}}{\left(\left(x^{4}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-20.136564990.63402993
-1-1.5574077-2.8041121
0.58.7408398-49.151576
11.5574077-2.8041121
2-0.136564990.63402993
3-0.00175983390.014942982

Questions

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

The derivative of tan⁡(x)x4\frac{\tan\left(x\right)}{x^{4}} is x4cos⁡(x)2−tan⁡(x)4x3(x4)2\frac{\frac{x^{4}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 4 x^{3}}{\left(x^{4}\right)^{2}}.

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

(((4 * x3x^{3} * cos(x)^2 - x4x^{4} * 2cos(x) * -sin(x)) / (cos(x)^2)^2 - (4 * x3x^{3} / cos(x)^2 + tan(x) * 12 * x2x^{2})) * (x4x^{4})^2 - (x4x^{4} / cos(x)^2 - tan(x) * 4 * x3x^{3}) * 2 * x4x^{4} * 4 * x3x^{3}) / ((x4x^{4})^2)^2.

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

The quotient rule.

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

-2.8041121.

Related derivatives

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