Derivative Problem - General Confusion

[AustrianSaurkraut]

Find dy/dx for the function e^y=5^x+y

So for this question, I'm confused as to where to start. Should I use ln to bring down y but then what do I do with the x+y? Any help is greatly appreciated as I have been struggling with this problem for a few days.

 

[nasi112]

What about implicit differentiation?

 

[JeffM]

As nasi says you could use implicit differentiation, but you could also do this:

[MATH]e^y = 5^{(x + y)} = 5^x * 5^y \implies ln(e^y) = ln(5^x * 5^y).[/MATH]
Now what?

 

[AustrianSaurkraut]

What about implicit differentiation?

I'm unsure what exactly you mean by implicit differentiation, should I do something like this:

ln(e^y) = ln(5^x+y)
y = x+y * ln(5)

It is at this step that I begin to get confused, as I don't really know where to go from here.

 

[JeffM]

I'm unsure what exactly you mean by implicit differentiation, should I do something like this:

ln(e^y) = ln(5^x+y)
y = x+y * ln(5)

It is at this step that I begin to get confused, as I don't really know where to go from here.

Note that your second line is in error.

[MATH]e^y = 5^{(x+y)} \implies ln(e^y) = ln(5^{(x + y)}) \implies y * ln(e) = (x + y) * ln(5) \implies y = x * ln(5) + y * ln(5).[/MATH]
Can you solve for y? Maybe by moving all the terms in y to the same side of the equation? You cannot forget basic algebra just because you are studying calculus.

 

[nasi112]

I'm unsure what exactly you mean by implicit differentiation, should I do something like this:

ln(e^y) = ln(5^x+y)
y = x+y * ln(5)

It is at this step that I begin to get confused, as I don't really know where to go from here.

implicit differentiation means differentiate both sides with respect to x without the need to isolate [MATH]y[/MATH]