Find the derivative of sec x

We need to find the derivative of sec x

Solution

Let us assume that sec x= 1/cos x

Now will use the quotient rule

{The quotient rule of differentiation is defined as the ratio of two functions (1st function / 2nd Function), is equal to the ratio of (Differentiation of 1st function × the 2nd function – Differentiation of second function × the 1st function) to the square of the 2nd function.

f′(x)=[s(x) / t(x)]′=t(x).s′(x)–s(x).t′(x) / {t(x)}2 }

So by the above mentioned quotient rule

quotient rule derivative of secx = 1/cosx

= \(\frac{\frac{d}{dx}\times 1 – \frac{d}{dx} cos x \times 1}{\left ( cosx \right )^{2}}\)

On simplifying the equation we get

= \(\frac{0 \times cosx – \frac{d}{dx}cos x \times 1}{\left ( cosx \right )^{2}}\)

= \(\frac{sin x}{\left ( cosx\times cosx \right )}\)

= \(\left ( \frac{sin x}{cosx} \right )(\frac{1}{cosx})\)

=tanx. secx

Answer

Derivative of sec x = tanx. secx

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