How to Find the Derivative of the Tangent Function and its Simplified Form

d/dx(tanx)

To find the derivative of the tangent function with respect to x, we can use the quotient rule

To find the derivative of the tangent function with respect to x, we can use the quotient rule.

The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x))/[h(x)]^2

In this case, g(x) = sin(x) and h(x) = cos(x).

So, we can start by finding the derivatives of g(x) and h(x).

g'(x) = d/dx(sin(x)) = cos(x)

h'(x) = d/dx(cos(x)) = -sin(x)

Now, substituting these derivatives into the quotient rule formula, we get:

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

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

Recall the identity cos^2(x) + sin^2(x) = 1, so we can simplify further:

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

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

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