e^x y =3

find dy/dx using implicit differentiation
the answer is -y

Compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative.
y=?
dy/dx=?
help me out with these last two please

I don't know what they are asking

y = 3e^-x

y' = -3e^-x

done implicitly:

e^x y = 3
e^x (dy/dx) + y e^x = 0
dy/dx = -y e^x/e^x = -y

solving for y first:
y = 3/e^x = 3 e^-x
dy/dx = 3(e^-x)(-1)
or
= -3/e^x

thank you that one was correct, i just had one other i couldn't get, the problem is find dy/dx using implicit differentiation: 4x+3y=xy

i found the derivative to be...
y-4/3-x

same question as the last one,
y=?
dy/dx=?

Thanks again for your help

i submitted the answer steve had first and it came out correct for the previous question

4x+3y=xy

4 + 3dy/dx = x(dy/dx) + y
3dy/dx - xdy/dx = y - 4
dy/dx(3-x) = y-4
dy/dx = (y-4)/(3-x) --- > you had that, if you had used the proper bracketing

solving the original for y ....

4x + 3y = xy
3y - xy = -4x
y(3-x) = -4x
y = -4x/(3-x) or 4x/(x-3)

now use quotient rule:

dy/dx = [4(x-3) - 4x(1) ]/(x-3)^2
= -12/(x-3)^2

Very nice, thanks Reiny

No problem, I can help you with that!

To find dy/dx using implicit differentiation, we differentiate both sides of the given equation with respect to x while treating y as an implicit function of x.

Let's differentiate both sides of the equation e^xy = 3 with respect to x:

d/dx (e^xy) = d/dx (3)

Next, we apply the chain rule on the left side:

(e^xy) * d/dx(xy) = 0

To differentiate xy with respect to x, we use the product rule:

y + x * dy/dx = 0

Now, we can isolate dy/dx by moving the other terms to the other side of the equation:

x * dy/dx = -y

Finally, we divide both sides of the equation by x to solve for dy/dx:

dy/dx = -y/x

So, using implicit differentiation, we found that dy/dx = -y/x.

To compare this result with finding y as a function of x and then taking the derivative, we can first solve the given equation e^xy = 3 for y as a function of x.

Taking the natural logarithm of both sides, we get:

ln(e^xy) = ln(3)

This simplifies to:

xy * ln(e) = ln(3)

Since ln(e) is equal to 1, the equation becomes:

xy = ln(3)

Solving for y, we divide both sides by x:

y = ln(3)/x

Now, we can find dy/dx by taking the derivative of this expression with respect to x.

Using the quotient rule, we find:

dy/dx = (1/x) * (-ln(3)/x^2)

This can be further simplified to:

dy/dx = -ln(3)/x^2

Comparing this result with our previous result from implicit differentiation (-y/x), we see that they are different:

dy/dx (implicit) = -y/x
dy/dx (explicit) = -ln(3)/x^2

So, the two methods yield different results.

Let me know if you have any further questions!