Question

a. use implicit differentiation to find $\frac{dy}{dx}$. b. find the slope of the curve at the given point. $y^{4}=4x; (64,4)$ a. $\frac{dy}{dx}=square$ b. the slope of the curve at $(64,4)$ is $square$. (simplify your answer.)

Explanation:

Step1: Differentiate both sides

Differentiate $y^{4}=4x$ with respect to $x$. Using the chain - rule on the left - hand side ($\frac{d}{dx}(y^{4}) = 4y^{3}\frac{dy}{dx}$) and the power - rule on the right - hand side ($\frac{d}{dx}(4x)=4$). So we have $4y^{3}\frac{dy}{dx}=4$.

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

Divide both sides of the equation $4y^{3}\frac{dy}{dx}=4$ by $4y^{3}$ to get $\frac{dy}{dx}=\frac{1}{y^{3}}$.

Step3: Find the slope at the given point

Substitute $x = 64$ and $y = 4$ into $\frac{dy}{dx}$. Since $\frac{dy}{dx}=\frac{1}{y^{3}}$, when $y = 4$, we have $\frac{dy}{dx}=\frac{1}{4^{3}}=\frac{1}{64}$.

Answer:

a. $\frac{dy}{dx}=\frac{1}{y^{3}}$
b. $\frac{1}{64}$