d/dx(tanx)
To find the derivative of the function f(x) = tan(x) with respect to x, we can use the chain rule
To find the derivative of the function f(x) = tan(x) with respect to x, we can use the chain rule.
The derivative of tan(x) can be obtained by differentiating the quotient sin(x)/cos(x).
Let’s break it down step by step:
1. Rewrite the function: tan(x) = sin(x)/cos(x).
2. Apply the quotient rule: The derivative of the quotient f(x) = g(x)/h(x) is given by (h(x)g'(x) – g(x)h'(x)) / [h(x)]^2.
3. Identify the numerator: g(x) = sin(x) and h(x) = cos(x).
4. Calculate the derivatives of g(x) and h(x): g'(x) = cos(x) and h'(x) = -sin(x).
5. Apply the quotient rule: f'(x) = [(cos(x) * cos(x)) – (sin(x) * (-sin(x)))] / [cos(x)]^2.
6. Simplify the numerator: f'(x) = (cos^2(x) + sin^2(x)) / [cos(x)]^2.
7. Use the identity sin^2(x) + cos^2(x) = 1: f'(x) = 1 / [cos(x)]^2.
8. Rewrite in terms of secant: Since sec^2(x) = 1 / [cos(x)]^2, we have f'(x) = 1 / sec^2(x).
Therefore, the derivative of tan(x) with respect to x is 1 / sec^2(x), which can be written as f'(x) = sec^2(x).
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