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Derivative of tan⁡(x)x3\frac{\tan\left(x\right)}{x^{3}}

The derivative of tan⁡(x)x3\frac{\tan\left(x\right)}{x^{3}} is x3cos⁡(x)2−tan⁡(x)3x2(x3)2\frac{\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}.

f(x) =
ddx[tan⁡(x)x3]=x3cos⁡(x)2−tan⁡(x)3x2(x3)2\frac{d}{dx}\left[\frac{\tan\left(x\right)}{x^{3}}\right] = \frac{\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}

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Graph of f(x) = tan(x) / x^3 and its derivative-3-2-1123-60-40-2020406080100120slope -1.247f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(x) / x^3 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -1.2467.

How to work it out

  1. 1

    We differentiate f(x)=tan⁡(x)x3f(x) = \frac{\tan\left(x\right)}{x^{3}} with respect to x. The rule that applies is the quotient rule.

    f(x)=tan⁡(x)x3f(x) = \frac{\tan\left(x\right)}{x^{3}}
  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=x3v = x^{3}.

    u=tan⁡(x),v=x3u = \tan\left(x\right),\quad v = x^{3}
  3. 3

    Differentiate each piece:

    u′=1cos⁡(x)2,v′=3x2u' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = 3 x^{2}
  4. 4

    Put them in the rule:

    f′(x)=(1cos⁡(x)2)(x3)−(tan⁡(x))(3x2)(x3)2f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(x^{3}\right) - \left(\tan\left(x\right)\right)\left(3 x^{2}\right)}{\left(x^{3}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=x3cos⁡(x)2−tan⁡(x)3x2(x3)2f'(x) = \frac{\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}
  6. 6

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

    f′′(x)=(3x2cos⁡(x)2−x32cos⁡(x)−sin⁡(x)(cos⁡(x)2)2−(3x2cos⁡(x)2+tan⁡(x)6x))(x3)2−(x3cos⁡(x)2−tan⁡(x)3x2)2x33x2((x3)2)2f''(x) = \frac{\left(\frac{3 x^{2} \cos\left(x\right)^{2} - x^{3} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{3 x^{2}}{\cos\left(x\right)^{2}} + \tan\left(x\right) 6 x\right)\right) \left(x^{3}\right)^{2} - \left(\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}\right) 2 x^{3} 3 x^{2}}{\left(\left(x^{3}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\tan\left(x\right)}{x^{3}}\right] = \frac{\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}} \]
\[ f(x) = \frac{\tan\left(x\right)}{x^{3}} \]
\[ u = \tan\left(x\right),\quad v = x^{3} \]
\[ u' = \frac{1}{\cos\left(x\right)^{2}},\quad v' = 3 x^{2} \]
\[ f'(x) = \frac{\left(\frac{1}{\cos\left(x\right)^{2}}\right)\left(x^{3}\right) - \left(\tan\left(x\right)\right)\left(3 x^{2}\right)}{\left(x^{3}\right)^{2}} \]
\[ f'(x) = \frac{\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{3 x^{2} \cos\left(x\right)^{2} - x^{3} 2 \cos\left(x\right) -\sin\left(x\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} - \left(\frac{3 x^{2}}{\cos\left(x\right)^{2}} + \tan\left(x\right) 6 x\right)\right) \left(x^{3}\right)^{2} - \left(\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}\right) 2 x^{3} 3 x^{2}}{\left(\left(x^{3}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-0.27312998-1.1314949
-11.55740771.2467044
0.54.3704199-15.834948
11.5574077-1.2467044
2-0.273129981.1314949
3-0.00527950160.043069113

Questions

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

The derivative of tan⁡(x)x3\frac{\tan\left(x\right)}{x^{3}} is x3cos⁡(x)2−tan⁡(x)3x2(x3)2\frac{\frac{x^{3}}{\cos\left(x\right)^{2}} - \tan\left(x\right) 3 x^{2}}{\left(x^{3}\right)^{2}}.

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

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

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

The quotient rule.

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

-1.2467044.

Related derivatives

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