Question

if (x^{2}+y^{2}=25), what is the value of (\frac{dy}{dx}) at the point ((4,3))?
a (-\frac{4}{3})
b (-\frac{3}{4})
c (\frac{3}{4})
d (\frac{4}{3})
e (\frac{25}{2})

Explanation:

Step1: Differentiate implicitly

Differentiate $x^{2}+y^{2}=25$ with respect to $x$. Using the power - rule and chain - rule, we get $2x + 2y\frac{dy}{dx}=0$.

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

Isolate $\frac{dy}{dx}$ in the equation $2x + 2y\frac{dy}{dx}=0$. First, subtract $2x$ from both sides: $2y\frac{dy}{dx}=-2x$. Then divide both sides by $2y$ (assuming $y
eq0$), so $\frac{dy}{dx}=-\frac{x}{y}$.

Step3: Substitute the point $(4,3)$

Substitute $x = 4$ and $y = 3$ into $\frac{dy}{dx}=-\frac{x}{y}$. We have $\frac{dy}{dx}=-\frac{4}{3}$.

Answer:

A. $-\frac{4}{3}$