Derivative of Tan(x): The Formula Explained and Derived

d/dx(tanx)

sec²x

To find the derivative of tan(x) with respect to x, we can use the definition of the tangent function and apply the chain rule. The derivative of tan(x) can be found as:

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

Now, applying the quotient rule:

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

Let’s find the derivatives of sin(x) and cos(x):

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

Substituting these values into the quotient rule formula:

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

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

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

Therefore, the derivative of tan(x) with respect to x is 1 / (cos(x))^2.

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