A Comprehensive Guide to Finding the Derivative of the Tangent Function Using the Quotient Rule

Derivative of tanx

To find the derivative of the tangent function, we need to use the quotient rule

To find the derivative of the tangent function, we need to use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x))/(h(x))^2

In the case of the tangent function, we can write it as tan(x) = sin(x)/cos(x). Now, let’s find the derivatives of sin(x) and cos(x):

Taking the derivative of sin(x):

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

Taking the derivative of cos(x):

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

Now, applying the quotient rule to find the derivative of tan(x):

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

Simplifying, we get:

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

As we know that sin^2(x) + cos^2(x) = 1, we can simplify further:

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

Therefore, the derivative of tan(x) with respect to x is:

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

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