Question

Determine derivatives (with respect to x) for the following g(x)=(5sqrtx +3x)^4

Answer 1

(d/dx)(5sqrtx+3x)^4 = d/dx((5x^(1/2)+3x)^4 =

4*((5x^(1/2)+3x))^3*d/dx((5x((1/2)+3x))) =

4((5x^(1/2)+3x))^3 *( ((5*1/2*x^(1/2-1)+3))=

4((5x^(1/2)+3x))^3*((5/2*x^(-1/2)+3)))=

4((5x^(1/2)+3x))^3*(((5/2*(1/x^(1/2) + 3))) =

4(5sqrtx+3x)^3 * ((( 5 /( 2sqrtx) +3)))

Comments

Why did the fourth power of the expression change to fourth root? I see that square root is the same as ^(1/2) but ^4 is not the same as ^(1/4). ??

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yep, I made a mistake

Answer 2

dg/dx=4(5sqrt(x)+3x)^3(5sqrt(x)/(2x)+3)

(differentiation by substitution: u=5sqrt(x)+3x, du/dx=5/(2sqrt(x))+3=5sqrt(x)/(2x)+3;

g(u)=u^4; dg/du=4u^3;

dg/dx=dg/du*du/dx=4u^3(5sqrt(x)/(2x)+3)=4(5sqrt(x)+3x)^3(5sqrt(x)/(2x)+3)).

Comments

du/dx = 5/2*x^(1/2-1)  + 3 =
5/2* x^(-1/2) + 3 =
(5/2)*1/sqrtx +3=
5/(2sqrtx) + 3,  not = to   5sqrtx/(2x) +3 why did u do this

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It's a form of rationalisation: 1/sqrt(x) is the same as sqrt(x)/x (multiply top and bottom of 1/sqrt(x) by sqrt(x): sqrt(x)/(sqrt(x))^2=sqrt(x)/x) so 5/(2sqrt(x))=5sqrt(x)/(2x). 1/sqrt(2) is usually represented as sqrt(2)/2. It just avoids putting square roots in denominators.