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Derivative of x5tan⁡(x)x^{5} \tan\left(x\right)

The derivative of x5tan⁡(x)x^{5} \tan\left(x\right) is 5x4tan⁡(x)+x51cos⁡(x)25 x^{4} \tan\left(x\right) + x^{5} \frac{1}{\cos\left(x\right)^{2}}.

f(x) =
ddx[x5tan⁡(x)]=5x4tan⁡(x)+x51cos⁡(x)2\frac{d}{dx}\left[x^{5} \tan\left(x\right)\right] = 5 x^{4} \tan\left(x\right) + x^{5} \frac{1}{\cos\left(x\right)^{2}}

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Graph of f(x) = x^5 * tan(x) and its derivative-3-2-1123-200-100100200slope 11.21f(x)f'(x)tangent at x = 1
Graph of f(x) = x^5 * tan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 11.213.

How to work it out

  1. 1

    We differentiate f(x)=x5tan⁡(x)f(x) = x^{5} \tan\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=x5tan⁡(x)f(x) = x^{5} \tan\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=x5u = x^{5} and v=tan⁡(x)v = \tan\left(x\right).

    u=x5,v=tan⁡(x)u = x^{5},\quad v = \tan\left(x\right)
  3. 3

    Differentiate each piece:

    u′=5x4,v′=1cos⁡(x)2u' = 5 x^{4},\quad v' = \frac{1}{\cos\left(x\right)^{2}}
  4. 4

    Put them in the rule:

    f′(x)=(5x4)(tan⁡(x))+(x5)(1cos⁡(x)2)f'(x) = \left(5 x^{4}\right)\left(\tan\left(x\right)\right) + \left(x^{5}\right)\left(\frac{1}{\cos\left(x\right)^{2}}\right)
  5. 5

    Simplify:

    f′(x)=5x4tan⁡(x)+x51cos⁡(x)2f'(x) = 5 x^{4} \tan\left(x\right) + x^{5} \frac{1}{\cos\left(x\right)^{2}}
  6. 6

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

    f′′(x)=20x3tan⁡(x)+5x41cos⁡(x)2+5x41cos⁡(x)2+x5−(2cos⁡(x)−sin⁡(x))(cos⁡(x)2)2f''(x) = 20 x^{3} \tan\left(x\right) + 5 x^{4} \frac{1}{\cos\left(x\right)^{2}} + 5 x^{4} \frac{1}{\cos\left(x\right)^{2}} + x^{5} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{5} \tan\left(x\right)\right] = 5 x^{4} \tan\left(x\right) + x^{5} \frac{1}{\cos\left(x\right)^{2}} \]
\[ f(x) = x^{5} \tan\left(x\right) \]
\[ u = x^{5},\quad v = \tan\left(x\right) \]
\[ u' = 5 x^{4},\quad v' = \frac{1}{\cos\left(x\right)^{2}} \]
\[ f'(x) = \left(5 x^{4}\right)\left(\tan\left(x\right)\right) + \left(x^{5}\right)\left(\frac{1}{\cos\left(x\right)^{2}}\right) \]
\[ f'(x) = 5 x^{4} \tan\left(x\right) + x^{5} \frac{1}{\cos\left(x\right)^{2}} \]
\[ f''(x) = 20 x^{3} \tan\left(x\right) + 5 x^{4} \frac{1}{\cos\left(x\right)^{2}} + 5 x^{4} \frac{1}{\cos\left(x\right)^{2}} + x^{5} \frac{-\left(2 \cos\left(x\right) -\sin\left(x\right)\right)}{\left(\cos\left(x\right)^{2}\right)^{2}} \]

Values of f and f' for x5x^{5} * tan(x)

xf(x)f'(x)
-2-69.921276-9.9775855
-11.5574077-11.212557
0.50.0170719530.21129598
11.557407711.212557
2-69.9212769.9775855
3-34.63881190.20629

Questions

What is the derivative of x5x^{5} * tan(x)?

The derivative of x5tan⁡(x)x^{5} \tan\left(x\right) is 5x4tan⁡(x)+x51cos⁡(x)25 x^{4} \tan\left(x\right) + x^{5} \frac{1}{\cos\left(x\right)^{2}}.

What is the second derivative of x5x^{5} * tan(x)?

20 * x3x^{3} * tan(x) + 5 * x4x^{4} * 1 / cos(x)^2 + 5 * x4x^{4} * 1 / cos(x)^2 + x5x^{5} * -(2cos(x) * -sin(x)) / (cos(x)^2)^2.

Which rule do you use for x5x^{5} * tan(x)?

The product rule.

What is the slope of x5x^{5} * tan(x) at x = 1?

11.212557.

Related derivatives

    Derivative of x^5 * tan(x): 5 * x^4 * tan(x) + x^5 * 1 / cos(x)^2 (with steps) | WordGPT