f'(tan⁻¹x)
To find the derivative of f(tan⁻¹x), we can use the chain rule
To find the derivative of f(tan⁻¹x), we can use the chain rule. Let’s break down the steps:
Step 1: Determine the composition of functions.
In this case, we have the function f(θ) composed with the function tan⁻¹(x). Therefore, we can write the composition as f(g(x)), where g(x) = tan⁻¹(x).
Step 2: Find the derivative of the inner function.
To find the derivative of g(x) = tan⁻¹(x), we differentiate it with respect to x. Recall that the derivative of tan⁻¹(x) is 1 / (1 + x²). Therefore, g'(x) = 1 / (1 + x²).
Step 3: Apply the chain rule.
The chain rule states that if we have a composition f(g(x)), the derivative is given by f'(g(x)) * g'(x).
For our case, since f(g(x)) = f(tan⁻¹(x)), the derivative of f(tan⁻¹(x)) is f'(tan⁻¹(x)) * g'(x).
Therefore, f'(tan⁻¹(x)) = f'(g(x)) * g'(x) = f'(θ) * (1 / (1 + x²)).
So, the derivative of f(tan⁻¹x) is f'(θ) / (1 + x²).
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