Question

(a) Find the slope of the curve \( ay^2 = x^3 \) at the point \( (am^2, am^3) \).

Hint:

Substitute the given point into the derivative after differentiating implicitly.

Answer 1

Differentiating\(ay^2=x^3\)withrespectto\(x\): \[ 2ay\frac{dy}{dx}=3x^2. \] At\((am^2,am^3)\),substitute\(x=am^2\)and\(y=am^3\): \[ 2a(am^3)\frac{dy}{dx}=3(am^2)^2. \] \[ 2am^3\frac{dy}{dx}=3a^2m^4. \] \[ \frac{dy}{dx}=\frac{3am^4}{2m^3}=\frac{3am}{2}. \]