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differentiate with respect to x from the first principles: sinx^cosx log10 to the base 'a' where a =x^x

differentiate with respect to x from the first principles:


sinx^cosx


log10 to the base 'a' where a =x^x

Grade:11

2 Answers

Askiitians_Expert Yagyadutt
askIITians Faculty 74 Points
11 years ago

Hiii !varsha

 

1) sinx ^ cosx

 

Let  it as  y = sinx ^ cos x

 

ln (y) = cosx . ln(sinx)

 

1/y .dy/dx = -sinx.ln(sinx) + cos^2(x)/sinx

 

dy/dx = y [ cos^2(x)/sin(x) - sin(x).ln(sinx) ]

 

Substitute y = sinx^cosx

 

2)  log 10 base a  = y

 

ln(10)/ln(a) = y

1/y = ln(a)/ln(10)

 

1/y = ln(x^x)/ln(10)  = x.ln(x)/ln10

 

(-1/y^2) dy.dx = 1/ln(10) + ln(x)/10

 

dy/dx = -y^2 ( ln(x) + 1)/ln(10)

 

SUbstitute y = ln(10)/ln(x^x)

 

 

With regards

Yagya

askiitians_expert

varsha rao
29 Points
11 years ago

Can you please do it from the first principles.

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