If #f(x)= 2 x^2 + x # and #g(x) = sqrtx + 1 #, how do you differentiate #f(g(x)) # using the chain rule?

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Answer 1 · 2016-01-11T19:45:31

#2 +5/(2x^(1/2) #

find f(g(x)) = f(#sqrtx + 1 ) = 2(sqrtx + 1 )^2 + sqrtx + 1 #

now square out the brackets and collect like terms.

noting that #(sqrtx)^2 = x #

#2(sqrtx + 1 )^2 + sqrtx + 1 = 2( x + 2sqrtx + 1 ) + sqrtx + 1 #

#rArr f(g(x)) = 2x + 4sqrtx + 2 + sqrtx + 1 = 2x + 5sqrtx + 3 #

now rewrite # sqrtx = x^(1/2) #

then f(g(x)) = #2x +5x^(1/2) + 3 #

differentiating to obtain.

f'(g(x)) = #2 + 5 xx1/2 x^(-1/2) = 2 + 5/(2x^(1/2) #