Porisma

Worked solution

Implicit differentiation with an exponential

A curve has equation xy2+3ey=4xxy^2+3e^y=4x. Find dydx\dfrac{dy}{dx} in terms of xx and yy.

[2008/MJC/Promo/10]

CJC · Discussion · Question 3

Answer

dydx=4−y22xy+3ey\dfrac{dy}{dx}=\dfrac{4-y^2}{2xy+3e^y}.

Differentiate

Product and chain rules give x⋅2ydydx+y2+3eydydx=4x\cdot2y\dfrac{dy}{dx}+y^2+3e^y\dfrac{dy}{dx}=4. Collect the derivative terms: (2xy+3ey)dydx=4−y2(2xy+3e^y)\dfrac{dy}{dx}=4-y^2. Therefore dydx=4−y22xy+3ey\dfrac{dy}{dx}=\dfrac{4-y^2}{2xy+3e^y} where the denominator is nonzero.

How did this part go?

Common mistakes

The derivative of xy² includes the y² term from differentiating x.

Read the hint