compute $\frac{dy}{dx}$ for $y = u^{3/2}$ and $u = 2x + 5$ using the chain - rule formula $\frac{dy}{dx}=\fra…

Explanation:

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

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

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

Differentiate $u = 2x + 5$ with respect to $x$. Since $\frac{d}{dx}(ax + b)=a$ (where $a = 2$ and $b = 5$), we get $\frac{du}{dx}=2$.

Step3: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{3}{2}u^{\frac{1}{2}}$ and $\frac{du}{dx}=2$ into the formula: $\frac{dy}{dx}=\frac{3}{2}u^{\frac{1}{2}}\cdot2 = 3u^{\frac{1}{2}}$.

Step4: Substitute $u = 2x+5$ back in

Replace $u$ with $2x + 5$ in the expression for $\frac{dy}{dx}$. So $\frac{dy}{dx}=3(2x + 5)^{\frac{1}{2}}$.

Answer:

$3(2x + 5)^{\frac{1}{2}}$