Derivative of \(tanx = sec^2x \)
What Is The Derivative Of tan(x)?
\( \frac{d}{dx} {tanx} = \frac{d}{dx} \frac{sinx}{cosx}\) [we know that \( tanx =\frac{sinx}{cosx} \)]
\( => \frac{d}{dx} {tanx} = \frac{cosx \times \frac{d}{dx} {sinx} – \frac{d}{dx} {cosx} \times sinx}{cos^2x} \) [Use Quotient Rule]
\( => \frac{d}{dx} {tanx} = \frac{cosx \times cosx – (-sinx) \times sinx}{cos^2x} \) [Simply]
\( => \frac{d}{dx} {tanx} = \frac{cos^2x + sin^2x }{cos^2x} \)
Use the Pythagorean identity for sine and cosine [\( sin^2x + cos^2x =1 \)]
\( => \frac{d}{dx} {tanx} = \frac{1 }{cos^2x} \)
we know that \( \frac{1}{cosx} =secx \)
\( => \frac{d}{dx} {tanx} = sec^2x \)
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