Learn To Find Derivative Of Tan(X) With Quotient Rule – Step-By-Step Guide

d/dx(tanx)

sec²x

To find the derivative of tan(x) with respect to x, we use the quotient rule.

Let f(x) = sin(x) and g(x) = cos(x), then tan(x) = f(x) / g(x).

Using the quotient rule:

[d/dx (tan(x))] = [ g(x) d/dx(f(x)) – f(x) d/dx(g(x)) ] / [g(x)]^2

Now, we find the derivatives of f(x) and g(x):

d/dx (sin(x)) = cos(x) and d/dx (cos(x)) = -sin(x)

Substituting these values in the above equation, we get:

[d/dx (tan(x))] = [cos(x)cos(x) – sin(x)(-sin(x))] / [cos(x)]^2

Simplifying the expression, we get:

[d/dx (tan(x))] = [cos^2(x) + sin^2(x)] / [cos^2(x)]

[d/dx (tan(x))] = 1 / [cos^2(x)]

Thus, the derivative of tan(x) with respect to x is (1 / cos^2(x)).

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