Question

f(x)=4^sqrt(x)

f'(x)= ?

please show steps to getting the answer.

Answer 1

Sqrt of 4 = 2

f(x) = 2x

Answer 2

Substitution:

f( x ) = f ( u ) = 4ᵘ

where

u = √ x

Apply the chain rule:

d f (u ) / dx = d f (u ) / du ∙ du / dx = d( 4ᵘ ) / du ∙ d( √ x ) / dx

d( u ) / du = d( 4ᵘ ) / du

Apply the derivative exponent rule:

d( aⁿ ) / du = aⁿ ∙ ln ( a )

d( aᵘ ) / du = aᵘ ∙ ln ( a )

d( √ x ) / dx

Apply the Power rule:

√ x = x^(1/2)

d( xⁿ ) / dx = n ∙ xⁿ⁻¹

d( x^( 1/2) ) / dx = 1/2 ∙ ( x^( 1/2 - 1 ) = 1/2 ∙ ( x^( - 1/2 ) = 1 / 2 ∙ x^(1/2 ) = 1 / 2√ x

d( u ) / dx = d( 4ᵘ ) / du ∙ d( √ x ) / dx =

4^√x ∙ ln ( 4 ) ∙ 1 / 2√ x = 4^√x ∙ ln ( 4 ) / 2√ x

Since:

4 = 2²

ln ( 4 ) = 2 ln ( 2 )


d( 4^√x ) / dx= 4^√x ∙ 2 ln ( 2 ) / 2√ x = 4^√x ∙ ln ( 2 ) / √ x