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

The derivative of tan⁡(x3)\tan\left(x^{3}\right) is 3x2cos⁡(x3)2\frac{3 x^{2}}{\cos\left(x^{3}\right)^{2}}.

f(x) =
ddx[tan⁡(x3)]=3x2cos⁡(x3)2\frac{d}{dx}\left[\tan\left(x^{3}\right)\right] = \frac{3 x^{2}}{\cos\left(x^{3}\right)^{2}}

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Graph of f(x) = tan(x^3) and its derivative-3-2-1123100200300400slope 10.28f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(x^3) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 10.277.

How to work it out

  1. 1

    We differentiate f(x)=tan⁡(x3)f(x) = \tan\left(x^{3}\right) with respect to x. The rule that applies is the chain rule.

    f(x)=tan⁡(x3)f(x) = \tan\left(x^{3}\right)
  2. 2

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=x3g(x) = x^{3}.

    g(x)=x3,g′(x)=3x2g(x) = x^{3},\quad g'(x) = 3 x^{2}
  3. 3

    The outer function is tan(u); its derivative evaluated at g(x), times g'(x):

    f′(x)=[1cos⁡(u)2]u=g(x)⋅g′(x)f'(x) = \left[\frac{1}{\cos\left(u\right)^{2}}\right]_{u=g(x)} \cdot g'(x)
  4. 4

    Simplify:

    f′(x)=3x2cos⁡(x3)2f'(x) = \frac{3 x^{2}}{\cos\left(x^{3}\right)^{2}}
  5. 5

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

    f′′(x)=6xcos⁡(x3)2−3x22cos⁡(x3)−(sin⁡(x3)3x2)(cos⁡(x3)2)2f''(x) = \frac{6 x \cos\left(x^{3}\right)^{2} - 3 x^{2} 2 \cos\left(x^{3}\right) -\left(\sin\left(x^{3}\right) 3 x^{2}\right)}{\left(\cos\left(x^{3}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\tan\left(x^{3}\right)\right] = \frac{3 x^{2}}{\cos\left(x^{3}\right)^{2}} \]
\[ f(x) = \tan\left(x^{3}\right) \]
\[ g(x) = x^{3},\quad g'(x) = 3 x^{2} \]
\[ f'(x) = \left[\frac{1}{\cos\left(u\right)^{2}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{3 x^{2}}{\cos\left(x^{3}\right)^{2}} \]
\[ f''(x) = \frac{6 x \cos\left(x^{3}\right)^{2} - 3 x^{2} 2 \cos\left(x^{3}\right) -\left(\sin\left(x^{3}\right) 3 x^{2}\right)}{\left(\cos\left(x^{3}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-26.7997115566.83291
-1-1.557407710.276556
0.50.125655140.76184191
11.557407710.276556
2-6.7997115566.83291
3-3.2737038316.36269

Questions

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

The derivative of tan⁡(x3)\tan\left(x^{3}\right) is 3x2cos⁡(x3)2\frac{3 x^{2}}{\cos\left(x^{3}\right)^{2}}.

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

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

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

The chain rule.

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

10.276556.

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