How to Find the Derivative of y = tan(x) Using the Chain Rule

dy/dx tanx

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

To find the derivative of y = tan(x) with respect to x, we can use the chain rule. First, let’s write tan(x) in terms of sine and cosine:

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

Now, we can differentiate both sides of this equation with respect to x. Remember that the derivative of sine is cosine and the derivative of cosine is negative sine:

d/dx(tan(x)) = d/dx(sin(x) / cos(x))
= (cos(x) * cos(x) – (-sin(x)) * sin(x)) / cos^2(x)
= (cos^2(x) + sin^2(x)) / cos^2(x)
= 1 / cos^2(x)

However, we can also express cos^2(x) in terms of tan(x) using the identity: 1 + tan^2(x) = sec^2(x). Rearranging this equation, we get:

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

Substituting this back into our previous result:

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

So, the derivative of y = tan(x) with respect to x is equal to 1 + tan^2(x).

Note: It’s important to be careful with the domain of tan(x) as it is periodic with intervals of π and undefined at odd multiples of π/2.

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