## derv tan(x)

### To find the derivative of the tangent function, we can use the quotient rule

To find the derivative of the tangent function, we can use the quotient rule. Let’s start with the equation:

f(x) = tan(x)

First, let’s express the tangent function as the ratio of sine and cosine:

f(x) = sin(x) / cos(x)

Now, using the quotient rule, we can find the derivative:

f'(x) = [cos(x) * d(sin(x))/dx – sin(x) * d(cos(x))/dx] / [cos(x)]^2

To find the derivatives of sine and cosine, we’ll need to use the chain rule. The derivatives are:

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

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

Plugging these derivatives into the quotient rule, we get:

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

Simplifying, we have:

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

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

f'(x) = 1 / [cos(x)]^2

And this is the derivative of the tangent function, tan(x). The derivative shows that the rate of change of tangent with respect to x is inversely proportional to the square of the cosine of x.

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