Question:
(a) Find the slope of the curve \( ay^2 = x^3 \) at the point \( (am^2, am^3) \).
(a) Find the slope of the curve \( ay^2 = x^3 \) at the point \( (am^2, am^3) \).
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Substitute the given point into the derivative after differentiating implicitly.
Updated On: Mar 1, 2025
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Solution and Explanation
Differentiating\(ay^2=x^3\)withrespectto\(x\):
\[
2ay\frac{dy}{dx}=3x^2.
\]
At\((am^2,am^3)\),substitute\(x=am^2\)and\(y=am^3\):
\[
2a(am^3)\frac{dy}{dx}=3(am^2)^2.
\]
\[
2am^3\frac{dy}{dx}=3a^2m^4.
\]
\[
\frac{dy}{dx}=\frac{3am^4}{2m^3}=\frac{3am}{2}.
\]
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