derivative of tan⁻¹x
derivative of tan⁻¹x is 1 / (1 + x²).
The derivative of tan⁻¹x can be found using the formula for the derivative of inverse functions:
If f(x) = y and g(y) = x, then (g o f)(x) = x and (f o g)(y) = y.
If f(x) = tan⁻¹x, then tan(f(x)) = x. Taking the derivative of both sides with respect to x, we get:
sec²(f(x)) * f'(x) = 1
Simplifying the expression using the identity 1 + tan²(f(x)) = sec²(f(x)), we get:
f'(x) = 1 / (1 + x²)
Therefore, the derivative of tan⁻¹x is 1 / (1 + x²).
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