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Derivative of tan⁡(x2)\tan\left(x^{2}\right)

The derivative of tan⁡(x2)\tan\left(x^{2}\right) is 2xcos⁡(x2)2\frac{2 x}{\cos\left(x^{2}\right)^{2}}.

f(x) =
ddx[tan⁡(x2)]=2xcos⁡(x2)2\frac{d}{dx}\left[\tan\left(x^{2}\right)\right] = \frac{2 x}{\cos\left(x^{2}\right)^{2}}

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Graph of f(x) = tan(x^2) and its derivative-3-2-1123-40-202040slope 6.851f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(x^2) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 6.851.

How to work it out

  1. 1

    We differentiate f(x)=tan⁡(x2)f(x) = \tan\left(x^{2}\right) with respect to x. The rule that applies is the chain rule.

    f(x)=tan⁡(x2)f(x) = \tan\left(x^{2}\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)=x2g(x) = x^{2}.

    g(x)=x2,g′(x)=2xg(x) = x^{2},\quad g'(x) = 2 x
  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)=2xcos⁡(x2)2f'(x) = \frac{2 x}{\cos\left(x^{2}\right)^{2}}
  5. 5

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

    f′′(x)=2cos⁡(x2)2−2x2cos⁡(x2)−(sin⁡(x2)2x)(cos⁡(x2)2)2f''(x) = \frac{2 \cos\left(x^{2}\right)^{2} - 2 x 2 \cos\left(x^{2}\right) -\left(\sin\left(x^{2}\right) 2 x\right)}{\left(\cos\left(x^{2}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\tan\left(x^{2}\right)\right] = \frac{2 x}{\cos\left(x^{2}\right)^{2}} \]
\[ f(x) = \tan\left(x^{2}\right) \]
\[ g(x) = x^{2},\quad g'(x) = 2 x \]
\[ f'(x) = \left[\frac{1}{\cos\left(u\right)^{2}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{2 x}{\cos\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{2 \cos\left(x^{2}\right)^{2} - 2 x 2 \cos\left(x^{2}\right) -\left(\sin\left(x^{2}\right) 2 x\right)}{\left(\cos\left(x^{2}\right)^{2}\right)^{2}} \]

Values of f and f' for tan(x2x^{2})

xf(x)f'(x)
-21.1578213-9.3622005
-11.5574077-6.8510376
0.50.255341921.0651995
11.55740776.8510376
21.15782139.3622005
3-0.452315667.2275367

Questions

What is the derivative of tan(x2x^{2})?

The derivative of tan⁡(x2)\tan\left(x^{2}\right) is 2xcos⁡(x2)2\frac{2 x}{\cos\left(x^{2}\right)^{2}}.

What is the second derivative of tan(x2x^{2})?

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

Which rule do you use for tan(x2x^{2})?

The chain rule.

What is the slope of tan(x2x^{2}) at x = 1?

6.8510376.

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