Question

Find dy/dx by implicit differentiation and evaluate the derivative at the given point.
xy = 35

Answer 1

\[xy=35\]
First, we differentiate with respect to x. We have to use the product rule to do this.
\[x \frac{dy}{dx}+y \frac{dx}{dx}=0\]dx/dx is simply 1, so this simplifies to
\[x \frac{dy}{dx}+y=0\]
We are looking for the derivative of y with respect to x, so solve for dy/dx
\[\frac{dy}{dx}=-\frac{y}{x}\]
Go back to the original function. We can solve for y in terms of x: y=35/x. Substitute this for y to get a function that depends only on x:
\[\frac{dy}{dx}=-\frac{\frac{35}{x}}{x}=-\frac{35}{x^2}\]
So\[y'=-\frac{35}{x^2}\]

The second part of your question said to evaluate the derivative at a certain point. You should just be able to plug the x-coordinate of the point into this derivative function.

Answer 2

I never understand why they ask students to implicitly differentiate that which can be differentiated explicitly, I'm sure it just confuses them unnecessarily.

Answer 3

Agreed. I checked the answer I got by explicitly differentiating. It was also much faster.