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

The derivative of x4tan⁡(x)x^{4} \tan\left(x\right) is 4x3tan⁡(x)+x41cos⁡(x)24 x^{3} \tan\left(x\right) + x^{4} \frac{1}{\cos\left(x\right)^{2}}.

f(x) =
ddx[x4tan⁡(x)]=4x3tan⁡(x)+x41cos⁡(x)2\frac{d}{dx}\left[x^{4} \tan\left(x\right)\right] = 4 x^{3} \tan\left(x\right) + x^{4} \frac{1}{\cos\left(x\right)^{2}}

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Graph of f(x) = x^4 * tan(x) and its derivative-3-2-1123100200300400slope 9.655f(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) = 9.6551.

How to work it out

  1. 1

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

    f(x)=x4tan⁡(x)f(x) = x^{4} \tan\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. 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)f'(x) = \left(4 x^{3}\right)\left(\tan\left(x\right)\right) + \left(x^{4}\right)\left(\frac{1}{\cos\left(x\right)^{2}}\right)
  5. 5

    Simplify:

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

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

xf(x)f'(x)
-234.96063822.469112
-1-1.55740779.6551497
0.50.0341439060.35430415
11.55740779.6551497
2-34.96063822.469112
3-11.5462767.250854

Questions

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

The derivative of x4tan⁡(x)x^{4} \tan\left(x\right) is 4x3tan⁡(x)+x41cos⁡(x)24 x^{3} \tan\left(x\right) + x^{4} \frac{1}{\cos\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.

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

The product rule.

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

9.6551497.

Related derivatives

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