differentiate with respect to x from the first principles:sinx^cosxlog10 to the base 'a' where a =x^x

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

Can you please do it from the first principles.