`y = 2x log_10 (sqrt(x))` Differentiate the function.

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Chapter 3, 3.6 - Problem 21 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sbernabel | Student, Graduate | (Level 1) Adjunct Educator

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Given `y=2*x*log10(sqrt(x))` we wish to find y'(x).

Write y into separate functions as follows:

let `f(x)=2*x, g(x)=log10(x), w(x)=sqrt(x).`

Use the Chain rule: `d/dx F(G(x))= F'(G(x)) * G'(x)`

and the Product rule: `d/dx (F(x)G(x)) = F'(x)G(x) + F(x)*G'(x)`

Compute `f'(x), g'(x) and w'(x)` ,

`f'(x)=2.`

Note that `g(x)=log10(x)=ln(x)/ln(10)=1/ln(10) * ln(x)`

so that `g'(x) = 1/ln(10) * 1/x.`

`w'(x) = 1/(2*sqrt(x)).`

Now apply the Chain rule and product rule:

`y'(x) = 2*log10(sqrt(x)) + 2*x*1/ln(10)*1/sqrt(x)*1/(2*sqrt(x))`

Combine like terms,

`y'(x) = log10(x) + 1/ln(10) = (ln(x)+1)/ln(10)`

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