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Derivative of tan⁡(3x)\tan\left(3 x\right)

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

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

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Graph of f(x) = tan(3x) and its derivative-3-2-112350100150200slope 3.061f(x)f'(x)tangent at x = 1
Graph of f(x) = tan(3x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.061.

How to work it out

  1. 1

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

    f(x)=tan⁡(3x)f(x) = \tan\left(3 x\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)=3xg(x) = 3 x.

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

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

    f′′(x)=−(32cos⁡(3x)−(3sin⁡(3x)))(cos⁡(3x)2)2f''(x) = \frac{-\left(3 2 \cos\left(3 x\right) -\left(3 \sin\left(3 x\right)\right)\right)}{\left(\cos\left(3 x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\tan\left(3 x\right)\right] = \frac{3}{\cos\left(3 x\right)^{2}} \]
\[ f(x) = \tan\left(3 x\right) \]
\[ g(x) = 3 x,\quad g'(x) = 3 \]
\[ f'(x) = \left[\frac{1}{\cos\left(u\right)^{2}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{3}{\cos\left(3 x\right)^{2}} \]
\[ f''(x) = \frac{-\left(3 2 \cos\left(3 x\right) -\left(3 \sin\left(3 x\right)\right)\right)}{\left(\cos\left(3 x\right)^{2}\right)^{2}} \]

Values of f and f' for tan(3x)

xf(x)f'(x)
-20.291006193.2540538
-10.142546543.0609586
0.514.10142599.55013
1-0.142546543.0609586
2-0.291006193.2540538
3-0.452315663.6137684

Questions

What is the derivative of tan(3x)?

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

What is the second derivative of tan(3x)?

-(3 * 2cos(3x) * -(3sin(3x))) / (cos(3x)^2)^2.

Which rule do you use for tan(3x)?

The chain rule.

What is the slope of tan(3x) at x = 1?

3.0609586.

Related derivatives

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