Applying implicit differentiation, it is found that:
\(\frac{d^2y}{dx^2}(4,3) = \frac{7}{27}\)
The curve given is:
\(x^2 + y^2 = 25\)
Applying implicit differentiation, we have that the first derivative is:
\(2x\frac{dx}{dx} + 2y\frac{dy}{dx} = 0\)
\(2y\frac{dy}{dx} = -2x\)
\(\frac{dy}{dx} = -\frac{x}{y}\)
Again applying implicit differentiation, the second derivative is:
\(\frac{d^2y}{dx^2} = -\frac{y\frac{dx}{dx} - x\frac{dy}{dx}}{y^2}\)
Since \(\frac{dy}{dx} = -\frac{x}{y}\)
\(\frac{d^2y}{dx^2} = -\frac{y - \frac{x^2}{y}}{y^2}\)
\(\frac{d^2y}{dx^2} = \frac{x^2 - y^2}{y^3}\)
At point (4,3), \(x = 4, y = 3\), then:
\(\frac{d^2y}{dx^2}(4,3) = \frac{4^2 - 3^2}{3^3} = \frac{7}{27}\)
A similar problem is given at https://brainly.com/question/15278071
