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Verdich [7]
3 years ago
5

Find dy/dx by implicit differentiation: x^2y^3-7xy^2=10

Mathematics
1 answer:
Ivenika [448]3 years ago
4 0

x^2y^3-7xy^2=10

Differentiate both sides wrt x:

\dfrac{\mathrm d(x^2y^3-7xy^2)}{\mathrm dx}=\dfrac{\mathrm d(10)}{\mathrm dx}

\dfrac{\mathrm d(x^2y^3)}{\mathrm dx}-7\dfrac{\mathrm d(xy^2)}{\mathrm dx}=0

By the product rule,

\left(y^3\dfrac{\mathrm d(x^2)}{\mathrm dx}+x^2\dfrac{\mathrm d(y^3)}{\mathrm dx}\right)-7\left(y^2\dfrac{\mathrm d(x)}{\mathrm dx}+x\dfrac{\mathrm d(y^2)}{\mathrm dx}\right)=0

By the power and chain rules,

\left(y^3(2x)+x^2\left(3y^2\dfrac{\mathrm dy}{\mathrm dx}\right)\right)-7\left(y^2+x\left(2y\dfrac{\mathrm dy}{\mathrm dx}\right)\right)=0

\left(2xy^3+3x^2y^2\dfrac{\mathrm dy}{\mathrm dx}\right)-7\left(y^2+2xy\dfrac{\mathrm dy}{\mathrm dx}\right)=0

(3x^2y^2-14xy)\dfrac{\mathrm dy}{\mathrm dx}=7y^2-2xy^3

\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{7y^2-2xy^3}{3x^2y^2-14xy}

and if y\neq0,

\implies\boxed{\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{7y-2xy^2}{3x^2y-14x}}

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