What is the implicit derivative of 1 = e x y ?

Question

What is the implicit derivative of       1    =          e              x        y            ?

helixl3u · Accepted Answer

The chain rule and the product rule must be combined when differentiating.
The left had side is a constant

1

so its derivative with respect to

x

is

0

For the right hand side we use the chain rule and the product rule.

e^{x y} [y + x \frac{d y}{d x}]

So together we have

0 = e^{x y} [y + x \frac{d y}{d x}]

Distribute

e^{x y}

0 = y e^{x y} + x e^{x y} \frac{d y}{d x}

Isolate term with

\frac{d y}{d x}

\frac{d y}{d x} x e^{x y} = - y e^{x y}

\frac{d y}{d x} = \frac{- y e^{x y}}{x e^{x y}}

\frac{d y}{d x} = - \frac{y}{x}

pritajena90k · Accepted Answer

1    =          e              x        y             implies that       x    y    =    0   which in turn implies that either       x    =    0   or       y    =    0  .The two axes that make up this equation&#039;s graph.Here is the graph of       1    =          e              x        y             using Socratic&#039;s graphing utility:graph{1=e^(xy) [-10, 10, -5, 5]}

What is the implicit derivative of 1=e^(xy)?

Answered question

Answer & Explanation

New Questions in Calculus 2