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
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
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
To find the derivative of the given function, we can use the product rule and the chain rule.
Let's break down the function into two parts: sqrt(x) and (x^3 + 1).
The first part can be represented as u = sqrt(x), and the second part can be represented as v = x^3 + 1.
Now, we can find the derivatives of u and v:
du/dx = (1/2) * (x^(-1/2)) (using power rule for derivatives)
dv/dx = 3x^2 (using power rule for derivatives)
Next, we can use the product rule to find the derivative of the entire function:
d/dx[u * v] = v * du/dx + u * dv/dx
= (x^3 + 1) * (1/2) * (x^(-1/2)) + sqrt(x) * 3x^2
Simplifying this expression gives us:
(1/2) * (x^3 + 1) * (x^(-1/2)) + 3x^2 * sqrt(x)
To simplify it further, we can rewrite x^(-1/2) as (1/sqrt(x)):
(1/2) * (x^3 + 1) * (1/sqrt(x)) + 3x^2 * sqrt(x)
This can be further simplified as:
(x^3 + 1) / (2 * sqrt(x)) + 3x^2 * sqrt(x)
So, the derivative of sqrt(x) * (x^3 + 1) is:
(7x^3 + 1) / (2 * sqrt(x))
Therefore, your answer is correct: (7x^3 + 1) / (2 * sqrt(x)).