How to Find the Derivative of the Tangent Function with the Chain Rule and Express It as sec^2(x)

d/dx[tan(x)]

To find the derivative of the tangent function with respect to x, we can use differentiation rules

To find the derivative of the tangent function with respect to x, we can use differentiation rules. In this case, we will use the chain rule.

The derivative of the tangent function can be written as:

d/dx[tan(x)] = sec^2(x)

Here’s a step-by-step explanation:

1. Start with the derivative of the tangent function:
d/dx[tan(x)]

2. Apply the chain rule:
If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

In our case, g(u) = tan(u) and h(x) = x. So, applying the chain rule:
d/dx[tan(x)] = d/du[tan(u)] * d/dx(x)

3. Differentiate the inner function u = x:
d/dx[tan(x)] = d/du[tan(u)] * 1

4. Find the derivative of tan(u):
Recall that the derivative of tan(u) is sec^2(u):
d/du[tan(u)] = sec^2(u)

5. Substitute back u = x:
d/dx[tan(x)] = sec^2(x)

So, the derivative of the tangent function with respect to x is sec^2(x).

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