Implicit Differentiation — Question 6

Question 6

Consider the equation defined implicitly: $$x^2 + y^2 + \ln(y) = 10$$

• Use implicit differentiation to find $\dfrac{dy}{dx}$.

• Find the slope of the tangent line at the point $(3, 1)$.

Question 6 - Solution

The logarithm requires $y>0$. Differentiating gives

$$2x+2yy'+y'/y=0,\qquad \boxed{y'=\frac{-2x}{2y+1/y}}.$$

At $(3,1)$, $3^2+1^2+\ln1=10$, so the point is on the curve.

The denominator is $3$, giving $\boxed{y'(3)=-2}$.