How to Find the Derivative of tan(x) Using the Quotient Rule and Identity

d/dx(tanx)

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

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

The derivative of tan(x) can be computed as follows:

Let’s start by rewriting tan(x) using the identity tan(x) = sin(x)/cos(x).

tan(x) = sin(x)/cos(x).

Now, applying the quotient rule, which states that the derivative of a quotient is given by:

d/dx(f(x)/g(x)) = (g(x)*f'(x) – f(x)*g'(x)) / [g(x)]^2,

where f(x) = sin(x) and g(x) = cos(x).

Differentiating f(x) and g(x), we have:

f'(x) = cos(x) (using the derivative of sin(x), which is cos(x))

g'(x) = -sin(x) (using the derivative of cos(x), which is -sin(x))

Now, plugging these values into the quotient rule formula:

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

Simplifying further:

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

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:

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

Alternatively, we can write this as:

d/dx(tan(x)) = sec^2(x),

where sec(x) is the reciprocal of cos(x). So, the derivative of tan(x) is equal to sec^2(x).

Therefore, the derivative of tan(x) with respect to x is sec^2(x).

More Answers: