WordGPT

Derivative of x4tan⁡(x)\frac{x^{4}}{\tan\left(x\right)}

The derivative of x4tan⁡(x)\frac{x^{4}}{\tan\left(x\right)} is 4x3tan⁡(x)−x41cos⁡(x)2tan⁡(x)2\frac{4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}}.

f(x) =
ddx[x4tan⁡(x)]=4x3tan⁡(x)−x41cos⁡(x)2tan⁡(x)2\frac{d}{dx}\left[\frac{x^{4}}{\tan\left(x\right)}\right] = \frac{4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}}{\tan\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^4 / tan(x) and its derivative-3-2-1123-600-500-400-300-200-100100slope 1.156f(x)f'(x)tangent at x = 1
Graph of f(x) = x^4 / tan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.1561.

How to work it out

  1. 1

    We differentiate f(x)=x4tan⁡(x)f(x) = \frac{x^{4}}{\tan\left(x\right)} with respect to x. The rule that applies is the quotient rule.

    f(x)=x4tan⁡(x)f(x) = \frac{x^{4}}{\tan\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=x4u = x^{4} and v=tan⁡(x)v = \tan\left(x\right).

    u=x4,v=tan⁡(x)u = x^{4},\quad v = \tan\left(x\right)
  3. 3

    Differentiate each piece:

    u′=4x3,v′=1cos⁡(x)2u' = 4 x^{3},\quad v' = \frac{1}{\cos\left(x\right)^{2}}
  4. 4

    Put them in the rule:

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

    Simplify:

    f′(x)=4x3tan⁡(x)−x41cos⁡(x)2tan⁡(x)2f'(x) = \frac{4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}}
  6. 6

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

    f′′(x)=(12x2tan⁡(x)+4x31cos⁡(x)2−(4x31cos⁡(x)2+x4−(2cos⁡(x)−sin⁡(x))(cos⁡(x)2)2))tan⁡(x)2−(4x3tan⁡(x)−x41cos⁡(x)2)2tan⁡(x)1cos⁡(x)2(tan⁡(x)2)2f''(x) = \frac{\left(12 x^{2} \tan\left(x\right) + 4 x^{3} \frac{1}{\cos\left(x\right)^{2}} - \left(4 x^{3} \frac{1}{\cos\left(x\right)^{2}} + x^{4} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}}\right)\right) \tan\left(x\right)^{2} - \left(4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}\right) 2 \tan\left(x\right) \frac{1}{\cos\left(x\right)^{2}}}{\left(\tan\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{4}}{\tan\left(x\right)}\right] = \frac{4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}} \]
\[ f(x) = \frac{x^{4}}{\tan\left(x\right)} \]
\[ u = x^{4},\quad v = \tan\left(x\right) \]
\[ u' = 4 x^{3},\quad v' = \frac{1}{\cos\left(x\right)^{2}} \]
\[ f'(x) = \frac{\left(4 x^{3}\right)\left(\tan\left(x\right)\right) - \left(x^{4}\right)\left(\frac{1}{\cos\left(x\right)^{2}}\right)}{\left(\tan\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(12 x^{2} \tan\left(x\right) + 4 x^{3} \frac{1}{\cos\left(x\right)^{2}} - \left(4 x^{3} \frac{1}{\cos\left(x\right)^{2}} + x^{4} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}}\right)\right) \tan\left(x\right)^{2} - \left(4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}\right) 2 \tan\left(x\right) \frac{1}{\cos\left(x\right)^{2}}}{\left(\tan\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-27.3225209-33.996249
-1-0.642092621.1560875
0.50.114405480.64332603
10.642092621.1560875
2-7.3225209-33.996249
3-568.23546-4824.9625

Questions

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

The derivative of x4tan⁡(x)\frac{x^{4}}{\tan\left(x\right)} is 4x3tan⁡(x)−x41cos⁡(x)2tan⁡(x)2\frac{4 x^{3} \tan\left(x\right) - x^{4} \frac{1}{\cos\left(x\right)^{2}}}{\tan\left(x\right)^{2}}.

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

((12 * x2x^{2} * tan(x) + 4 * x3x^{3} * 1 / cos(x)^2 - (4 * x3x^{3} * 1 / cos(x)^2 + x4x^{4} * -(2cos(x) * -sin(x)) / (cos(x)^2)^2)) * tan(x)^2 - (4 * x3x^{3} * tan(x) - x4x^{4} * 1 / cos(x)^2) * 2tan(x) * 1 / cos(x)^2) / (tan(x)^2)^2.

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

The quotient rule.

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

1.1560875.

Related derivatives

    Derivative of x^4 / tan(x): (4 * x^3 * tan(x) - x^4 * 1 / cos(x)^2) / tan(x)^2 (with steps) | WordGPT