Sqrt of 4 = 2
f(x) = 2x
f(x)=4^sqrt(x)
f'(x)= ?
please show steps to getting the answer.
2 answers
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
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