Learn How to Find the Derivative of tan(x) with Respect to x Using the Quotient Rule in Calculus

ddx tanx

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

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

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

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

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

We’ll start by finding f'(x) and g'(x):

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

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

Now, we can substitute these values into the quotient rule formula:

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

Simplifying further:

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

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

ddx tan(x) = 1 / cos^2(x)

Since cos^2(x) is equivalent to sec^2(x), we can rewrite the derivative as:

ddx tan(x) = 1 / sec^2(x)

And since sec^2(x) is equal to 1/cos^2(x), the derivative simplifies to:

ddx tan(x) = cos^2(x)

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

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