Differentiate each of the following?

Question

cos^8 2x

x.lnx

lnx/x

Answer 1

f (x) = cos (2x)^8

Use the chain rule:

dy/dx = d/dx ( cos (2x)^8 ) * d/dx ( cos (2x) ) * d/dx (2x)

dy/dx = 8cos (2x)^7 * -sin (2x) * 2

dy/dx = -16 cos (2x)^7 sin (2x)

f (x) = x ln (x)

Use the product rule:

dy/dx = d/dx (x) * ln (x) + (x) * d/dx ( ln (x) )

dy/dx = (1) ( ln (x) ) + (x) (1 / x)

dy/dx = ln (x) + 1

f (x) = ln (x) / x

Use the quotient rule:

dy/dx = [ d/dx ( ln (x) ) ( x ) - ( ln (x) ) * d/dx ( x ) ] / x²

dy/dx = [ (1/x) (x) - ( ln (x) (1) ] / x²

dy/dx = [ 1 - ln (x) ] / x²

Answer 2

1) 8cos^7 2x(-sin2x)(2) = -16cos^7 2x sin2x

2) lnx + 1

3) - 1/x^2 lnx + 1/x^2  

Answer 3

Question 1

f (x) = ( cos 2x )^8

f `(x) = 8 (cos 2x)^7 (- 2 sin 2x)

f `(x) = - 16 (cos 2x)^7 (sin 2x)

Question 2

f (x) = x ln x

f `(x) = ln x + (1/x) (x)

f `(x) = ln x + 1

Question 3

f `(x) = [ x (1/x) - ln x ]  / x²

f `(x) = [ 1 - ln x ]  / x²