d/dx [ tan(f(x)) ]
To find the derivative of tan(f(x)) with respect to x, we will use the chain rule
To find the derivative of tan(f(x)) with respect to x, we will use the chain rule. The chain rule states that if we have a function within another function, then the derivative of the composite function is the derivative of the outer function multiplied by the derivative of the inner function.
Let’s break down the problem step by step:
Step 1: Identify the inner and outer functions.
Here, the outer function is tan(u), where u = f(x). The inner function is f(x).
Step 2: Find the derivative of the inner function.
Let’s say the derivative of f(x) with respect to x is f'(x).
Step 3: Find the derivative of the outer function.
The derivative of tan(u) with respect to u is sec^2(u).
Step 4: Apply the chain rule using the derivatives from steps 2 and 3.
According to the chain rule, the derivative of tan(f(x)) with respect to x is:
d/dx[tan(f(x))] = d/dx[tan(u)] * du/dx
Substituting the derivatives from steps 2 and 3, we get:
d/dx[tan(f(x))] = sec^2(u) * f'(x)
Step 5: Substitute u = f(x).
Since u = f(x), we can replace u in the expression with f(x):
d/dx[tan(f(x))] = sec^2(f(x)) * f'(x)
Therefore, the derivative of tan(f(x)) with respect to x is sec^2(f(x)) times the derivative of f(x) with respect to x.
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