Mastering The Chain Rule: How To Find The Derivative Of Tan(X)

d/dx [ tan(x) ]

sec²(x)

To take the derivative of tan(x), we need to use the chain rule. The chain rule states that the derivative of a composition of functions is the product of the derivative of the outer function and the derivative of the inner function.

Let’s start by breaking down tan(x) as a composition of functions. We can write it as f(g(x)) where f(x) = tan(x) and g(x) = x.

f'(x) = sec^2(x) is the derivative of the outer function f(x) = tan(x).

g'(x) = 1 is the derivative of the inner function g(x) = x.

Using the chain rule, we get:

d/dx [tan(x)] = f'(g(x)) * g'(x) = sec^2(x) * 1 = sec^2(x)

Therefore, the derivative of tan(x) is sec^2(x).

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