a manufacturer can produce 4050 cell phones when $x$ dollars is spent on labor and $y$ dollars is spent on…

Explanation:

Step1: Differentiate both sides

Differentiate $75x^{\frac{3}{4}}y^{\frac{1}{4}} = 4050$ with respect to $x$ using the product - rule $(uv)^\prime=u^\prime v + uv^\prime$, where $u = 75x^{\frac{3}{4}}$ and $v=y^{\frac{1}{4}}$. The derivative of $u = 75x^{\frac{3}{4}}$ with respect to $x$ is $u^\prime=75\times\frac{3}{4}x^{-\frac{1}{4}}=\frac{225}{4}x^{-\frac{1}{4}}$, and the derivative of $v = y^{\frac{1}{4}}$ with respect to $x$ is $v^\prime=\frac{1}{4}y^{-\frac{3}{4}}\frac{dy}{dx}$. The derivative of the right - hand side (a constant 4050) with respect to $x$ is 0. So, $\frac{225}{4}x^{-\frac{1}{4}}y^{\frac{1}{4}}+75x^{\frac{3}{4}}\times\frac{1}{4}y^{-\frac{3}{4}}\frac{dy}{dx}=0$.

Step2: Solve for $\frac{dy}{dx}$

First, isolate the terms with $\frac{dy}{dx}$: $75x^{\frac{3}{4}}\times\frac{1}{4}y^{-\frac{3}{4}}\frac{dy}{dx}=-\frac{225}{4}x^{-\frac{1}{4}}y^{\frac{1}{4}}$. Then, multiply both sides by $\frac{4}{75x^{\frac{3}{4}}y^{-\frac{3}{4}}}$: $\frac{dy}{dx}=-\frac{225}{4}x^{-\frac{1}{4}}y^{\frac{1}{4}}\times\frac{4}{75x^{\frac{3}{4}}y^{-\frac{3}{4}}}$. Simplify the right - hand side: $\frac{dy}{dx}=-\frac{225}{75}\times\frac{y^{\frac{1}{4}+\frac{3}{4}}}{x^{\frac{1}{4}+\frac{3}{4}}}=-\frac{3y}{x}$.

Step3: Evaluate $\frac{dy}{dx}$ at the point $(81,16)$

Substitute $x = 81$ and $y = 16$ into $\frac{dy}{dx}=-\frac{3y}{x}$: $\frac{dy}{dx}=-\frac{3\times16}{81}=-\frac{48}{81}\approx - 0.5926$.

Answer:

a. $\frac{dy}{dx}=-\frac{3y}{x}$ b. $- 0.5926$