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derivative of tan^3(x)

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

The derivative of \tan^3(x) is 3\tan^2(x)\sec^2(x).

Solution. Let F'(x) = \tan^3(x) = f(g(x)), where f(u) = u^3 and g(x) = \tan(x). We will use the chain rule to determine the derivative of \tan^3(x):
F'(x) = f'(g(x))g'(x).
The derivative of f(u) = u^3 is f'(u) = 3u^2 and the derivative of g(x) = \tan(x) if g'(x) = \frac{1}{\cos^2(x)} which we have seen here. So we get:
f'(g(x)) = f'(\tan(x)) = 3\tan^2(x) \quad \text{and} \quad g'(x) = \frac{1}{\cos^2(x)}.
Substituting everything, we get:
F'(x) &= f'(g(x))g'(x) \\
&= 3\tan^2(x) \frac{1}{\cos^2(x)} \\
&= 3\tan^2(x)\sec^2(x),
since \frac{1}{\cos^2(x)} = \sec^2(x). Therefore, the derivative of \tan^3(x) is 3\tan^2(x)\sec^2(x).

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