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

Question

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

Samantha Adkison · Accepted Answer

#(dy)/dx=-y/x#

When we differentiate we have to use the chain rule in conjunction with the product rule.

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^(xy)[y+x(dy)/dx]#

So together we have

#0=e^(xy)[y+x(dy)/dx]#

Distribute #e^(xy)#

#0=ye^(xy)+xe^(xy)(dy)/dx#

Isolate term with #(dy)/dx#

#(dy)/dxxe^(xy)=-ye^(xy)#

#(dy)/dx=(-ye^(xy))/(xe^(xy))#

#(dy)/dx=-y/x#

Elijah Abram · Answer

I like the question and the answer can be written #dy/dx = y/x#, but . . .

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

Emma Aldridge · Answer

undefined To find the implicit derivative of $1 = e^{xy}$, differentiate both sides with respect to $x$:

$$
\frac{d}{dx}(1) = \frac{d}{dx}(e^{xy})
$$

The derivative of a constant is 0, and the derivative of $e^{xy}$ with respect to $x$ using the chain rule is:

$$
\frac{d}{dx}(e^{xy}) = e^{xy} \cdot \frac{d(xy)}{dx}
$$

Using the product rule for $xy$:

$$
\frac{d(xy)}{dx} = x \frac{dy}{dx} + y \frac{dx}{dx} = x \frac{dy}{dx} + y
$$

Substituting this into the equation:

$$
0 = e^{xy} \cdot (x \frac{dy}{dx} + y)
$$

Solving for $\frac{dy}{dx}$:

$$
\frac{dy}{dx} = -\frac{y}{x}
$$