How do you differentiate #f(x)= (4x^5+5)^(1/2)# using the chain rule?

2 Answers
Carl S.
May 22, 2018

#{d[f(x)]}/{dx} = {10x^4}/{sqrt{4x^5 + 5}}#

Explanation:

#f(x) = (4x^5 + 5)^{1/2}#

#{d[f(x)]}/{dx} = d/dx[(4x^5 + 5)^{1/2}]#

let #u = 4x^5 + 5#

#{d[f(x)]}/{dx} = d/{du}[u^{1/2}]d/dx[4x^5 + 5]#

#{d[f(x)]}/{dx} = 1/2u^{-1/2} times 20x^4#

Substitute #u# back in

#{d[f(x)]}/{dx} = 1/2( 4x^5 + 5)^{-1/2} times 20x^4#

Simplify

#{d[f(x)]}/{dx} = {10x^4}/{sqrt{4x^5 + 5}}#

Jim G.
May 22, 2018

#f'(x)=(10x^4)/(4x^5+5)^(1/2)#

Explanation:

#"given "f(x)=g(h(x))" then"#

#f'(x)=g'(h(x))xxh'(x)larrcolor(blue)"chain rule"#

#rArrf'(x)=1/2(4x^5+5)^(-1/2)xxd/dx(4x^5+5)#

#"color(white)(rArrf'(x))=1/cancel(2)(4x^5+5)^(-1/2)xxcancel(20)^(10)x^4#

#color(white)(rArrf'(x))=(10x^4)/(4x^5+5)^(1/2)#

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