Question

Find the derivative.

d/dx[sqrt of x (x^3 +1)]

My answer in its most simplified form:
(7x^3 + 1)/ 2 • sqrt of x

Answer 1

y = √x (x^3 + 1)
dy/dx = x(1/2) (3x^2) + (1/2)x^(-1/2) (x^3 + 1)
= (1/2)(x^(-1/2)) (6x^3 + x^3 + 1)
= (7x^3 + 1)/(2√x)

you are correct, but you need brackets around the denominator, or else it is only divided by 2

Answer 2

If this is y = [ x^4 + x ]^.5 ??????

dy/dx = .5 [ x^4 + x ]^-.5 (4x^3 +1)

= .5 (4x^3+1) /sqrt(x^4+x)

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if you mean
y =( x^.5 )(x^3+1)
y = x^3.5 + x^.5

dy/dx = 3.5 x^2.5 + .5 x^.5 /x

= x^.5 ( 7 x^2 + 1/x)/2

= x^.5 (7 x^3 + 1) /(2 x)

= (7 x^3 + 1) /[ 2 sqrt x ]

agree, but parentheses are IMPORTANT