Derivative of tan(x) with Respect to x: Finding the Derivative of the Tangent Function Using the Quotient Rule

To find the derivative of `tan(x)` with respect to `x`, we can use the quotient rule since `tan(x)` is the ratio of sine and cosine functions

To find the derivative of `tan(x)` with respect to `x`, we can use the quotient rule since `tan(x)` is the ratio of sine and cosine functions.

The quotient rule states that if we have a function of the form `f(x) = g(x) / h(x)`, then the derivative of `f(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)`. Let’s find the derivatives of `g(x)` and `h(x)` first.

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

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

Now, let’s substitute these values into the quotient rule formula:

f'(x) = [cos(x) * cos(x) – sin(x) * (-sin(x))] / [cos(x)]^2
= [cos^2(x) + sin^2(x)] / cos^2(x)
= 1 / cos^2(x)

However, we know that `tan(x)` can be expressed as the ratio of `sin(x)` and `cos(x)`, i.e., `tan(x) = sin(x) / cos(x)`.

So the derivative of `tan(x)` is equal to `1 / cos^2(x)`.

Thus, d/dx(tan(x)) = 1 / cos^2(x).

d/dx(tanx)