# Find the derivative of the following function: \[\sec x\]

Last updated date: 27th Mar 2023

•

Total views: 309k

•

Views today: 6.86k

Answer

Verified

309k+ views

Hint: To find the derivative of the given function, we will simplify the given function in terms of fractions using trigonometric relations and then find the derivative using quotient rule of differentiation.

We have the function \[y=\sec x\]. We have to find the first derivative of the given function.

Thus, we will differentiate the given function with respect to the variable \[x\].

We can rewrite \[y=\sec x\] in terms of \[\cos x\] as \[y=\sec x=\dfrac{1}{\cos x}\].

We will now use quotient rule to find the derivative of the given function which states that if \[y=\dfrac{f\left( x \right)}{g\left( x \right)}\], then we have \[\dfrac{dy}{dx}=\dfrac{g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right)}{{{g}^{2}}\left( x \right)}\].

We have to evaluate \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{dy}{dx}\left( \dfrac{1}{\cos x} \right)\].

Thus, substituting \[f\left( x \right)=1,g\left( x \right)=\cos x\] in the quotient rule of differentiation, we get \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{dy}{dx}\left( \dfrac{1}{\cos x} \right)=\dfrac{\cos x\times \dfrac{d}{dx}\left( 1 \right)-1\times \dfrac{d}{dx}\left( \cos x \right)}{{{\left( \cos x \right)}^{2}}}\]. \[...\left( 1 \right)\]

We know that differentiation of a constant is zero with respect to any variable. Thus, we have\[\dfrac{d}{dx}\left( 1 \right)=0\]. \[...\left( 2 \right)\]

We also know that differentiation of function of the form \[y=\cos x\] is \[\dfrac{dy}{dx}=-\sin x\]. \[...\left( 3 \right)\]

Substituting the value of equation \[\left( 2 \right), \left( 3 \right)\] in equation \[\left( 1 \right)\], we have \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{\cos x\times \dfrac{d}{dx}\left( 1 \right)-1\times \dfrac{d}{dx}\left( \cos x \right)}{{{\left( \cos x \right)}^{2}}}=\dfrac{\cos x\times 0-1\times \left( -\sin x \right)}{{{\cos }^{2}}x}=\dfrac{\sin x}{{{\cos }^{2}}x}\].

We know that \[\dfrac{\sin x}{\cos x}=\tan x\]. Thus, we have \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{\sin x}{{{\cos }^{2}}x}=\dfrac{\tan x}{\cos x}=\tan x\sec x\].

Hence, the derivative of the function \[y=\sec x\] is \[\dfrac{dy}{dx}\left( \sec x \right)=\tan x\sec x\].

The derivative of any function \[y=f\left( x \right)\] with respect to variable \[x\] is a measure of the rate at which the value of the function changes with respect to the change in the value of variable \[x\]. The first derivative of any function also signifies the slope of the function when the graph of \[y=f\left( x \right)\] is plotted against \[x\] considering only real values of the function.

Note: It’s necessary to use quotient rules to find the derivative of the given function. We can also use the basic formula for finding the derivative of any function using limit.

We have the function \[y=\sec x\]. We have to find the first derivative of the given function.

Thus, we will differentiate the given function with respect to the variable \[x\].

We can rewrite \[y=\sec x\] in terms of \[\cos x\] as \[y=\sec x=\dfrac{1}{\cos x}\].

We will now use quotient rule to find the derivative of the given function which states that if \[y=\dfrac{f\left( x \right)}{g\left( x \right)}\], then we have \[\dfrac{dy}{dx}=\dfrac{g\left( x \right)f'\left( x \right)-f\left( x \right)g'\left( x \right)}{{{g}^{2}}\left( x \right)}\].

We have to evaluate \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{dy}{dx}\left( \dfrac{1}{\cos x} \right)\].

Thus, substituting \[f\left( x \right)=1,g\left( x \right)=\cos x\] in the quotient rule of differentiation, we get \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{dy}{dx}\left( \dfrac{1}{\cos x} \right)=\dfrac{\cos x\times \dfrac{d}{dx}\left( 1 \right)-1\times \dfrac{d}{dx}\left( \cos x \right)}{{{\left( \cos x \right)}^{2}}}\]. \[...\left( 1 \right)\]

We know that differentiation of a constant is zero with respect to any variable. Thus, we have\[\dfrac{d}{dx}\left( 1 \right)=0\]. \[...\left( 2 \right)\]

We also know that differentiation of function of the form \[y=\cos x\] is \[\dfrac{dy}{dx}=-\sin x\]. \[...\left( 3 \right)\]

Substituting the value of equation \[\left( 2 \right), \left( 3 \right)\] in equation \[\left( 1 \right)\], we have \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{\cos x\times \dfrac{d}{dx}\left( 1 \right)-1\times \dfrac{d}{dx}\left( \cos x \right)}{{{\left( \cos x \right)}^{2}}}=\dfrac{\cos x\times 0-1\times \left( -\sin x \right)}{{{\cos }^{2}}x}=\dfrac{\sin x}{{{\cos }^{2}}x}\].

We know that \[\dfrac{\sin x}{\cos x}=\tan x\]. Thus, we have \[\dfrac{dy}{dx}\left( \sec x \right)=\dfrac{\sin x}{{{\cos }^{2}}x}=\dfrac{\tan x}{\cos x}=\tan x\sec x\].

Hence, the derivative of the function \[y=\sec x\] is \[\dfrac{dy}{dx}\left( \sec x \right)=\tan x\sec x\].

The derivative of any function \[y=f\left( x \right)\] with respect to variable \[x\] is a measure of the rate at which the value of the function changes with respect to the change in the value of variable \[x\]. The first derivative of any function also signifies the slope of the function when the graph of \[y=f\left( x \right)\] is plotted against \[x\] considering only real values of the function.

Note: It’s necessary to use quotient rules to find the derivative of the given function. We can also use the basic formula for finding the derivative of any function using limit.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE