Question

compute \\( \frac { d y } { d x } \\) for \\( y = u ^ { 3 / 2 } \\) and \\( u = 6 x + 5 \\) using the chain rule formula \\( \frac { d y } { d x } = \frac { d y } { d u } \frac { d u } { d x } \\)
begin by calculating \\( \frac { d y } { d u } \\) and \\( \frac { d u } { d x } \\).
\\( \frac { d y } { d u } = \square \\) (type an expression using \\( u \\) as the variable.)
\\( \frac { d u } { d x } = \square \\)

Explanation:

Step1: Calculate $\frac{dy}{du}$

Using the power rule $\frac{d}{du}(u^n)=nu^{n - 1}$, for $y = u^{\frac{3}{2}}$, we have $\frac{dy}{du}=\frac{3}{2}u^{\frac{3}{2}-1}=\frac{3}{2}u^{\frac{1}{2}}$.

Step2: Calculate $\frac{du}{dx}$

For $u = 6x+5$, using the power rule $\frac{d}{dx}(ax + b)=a$ (where $a = 6$ and $b = 5$), we get $\frac{du}{dx}=6$.

Answer:

$\frac{dy}{du}=\frac{3}{2}u^{\frac{1}{2}}$, $\frac{du}{dx}=6$