How To Find The Derivative Of Tanx Using The Quotient Rule: A Comprehensive Math Tutorial

derivative of tanx

secxtanx

The derivative of tanx is sec^2x.

Here’s how to find it:

Recall that tanx = sinx/cosx.

The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative is given by

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

Applying this rule to tanx, we have

tan'(x) = [sin'(x)cosx – sinx(cos'(x))] / [cosx]^2

Recall that sin'(x) = cosx and cos'(x) = -sinx.

Substituting these values, we get

tan'(x) = [cosx * cosx – sinx * (-sinx)] / [cosx]^2

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

Note that cos^2x + sin^2x = 1. Substituting this value, we get

tan'(x) = 1 / [cosx]^2

Since secx = 1/cosx, we can further simplify this to

tan'(x) = sec^2x

Therefore, the derivative of tanx is sec^2x.

More Answers: