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# Find the Derivative of the Following Function - CBSE (Commerce) Class 12 - Mathematics

ConceptDerivatives of Inverse Trigonometric Functions

#### Question

Find the derivative of the following function f(x) w.r.t. x, at x = 1 :

f(x)=cos^-1[sin sqrt((1+x)/2)]+x^x

#### Solution

f(x)=cos^-1[sin sqrt((1+x)/2)]+x^x

f1(x)                            f2(x)

Let  f_1(x)=cos^−1 [sin sqrt((1+x)/2)] and f_2(x)=x^x

Now,

f_1(x)=cos^−1 [sin sqrt((1+x)/2)]

= f_1(x)=cos^−1 [cos(pi/2- sqrt((1+x)/2))]

=pi/2- sqrt((1+x)/2)

⇒f_1'(x)=−1/2sqrt(2/(1+x))=−sqrt(1/(2(1+x)))

and

f_2(x)=x^x Taking log on both sides, we get

log f_2(x)=xlogx

⇒1/(f_2(x)) f2′(x)=logx+x⋅1/x

=>1/(f_2(x)) f2′(x)=logx+1

=>f2′(x)=f_2(x)(logx+1)

=>f2′(x)=x^x(logx+1)

∵f(x)=f_1(x)+f_2(x)

∵f'(x)=f_1'(x)+f_2'(x)

=-sqrt(1/(2(1+x)))+x^x (logx+1)

At x=1

f'(1)=-sqrt(1/(2(1+1)))1^1(log1+1)

=-1/2+1=1/2

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Solution Find the Derivative of the Following Function Concept: Derivatives of Inverse Trigonometric Functions.
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