How do you differentiate 4xy-3x-11=0?

2 Answers
Jul 16, 2015

dy/dx=(3-4y)/(4x)

Explanation:

Assume the equation 4xy-3x-11=0 implicitly defines y as a function of x and then differentiate with respect to x, using the Product Rule:

4y+4x dy/dx-3=0

Now solve for dy/dx to get dy/dx=(3-4y)/(4x).

If you happen to know a specific point (x,y) on the curve defined by the original equation 4xy-3x-11=0, you can plug the coordinates of that point into dy/dx=(3-4y)/(4x) to find the slope of the curve at that point.

The particular equation 4xy-3x-11=0 is simple enough that we can check our work another way; we can solve for y explictly as a function of x:

4xy-3x-11=0\Rightarrow y=(3x+11)/(4x)=3/4+11/4 x^{-1}

Now we can differentiate this in the ordinary way to get dy/dx=-11/4 x^{-2}=-11/(4x^2). Is this answer the same as the original? It sure looks different. It can be seen to be the same answer by substituting y=(3x+11)/(4x) into dy/dx=(3-4y)/(4x) in place of y and simplifying:

dy/dx=(3-4((3x+11)/(4x)))/(4x)=(3x-(3x+11))/(4x^2)=-11/(4x^2)

It is the same!

As an example of a point on this curve, we can use the equation y=(3x+11)/(4x) to find y when x=1 to be y=14/4=7/2, meaning that the point (x,y)=(1,7/2) is on the graph of 4xy-3x-11=0. The slope of the curve at that point is dy/dx=(3-14)/4=-11/4=-2.75.

The equation of the tangent line to the curve at that point is therefore y=-11/4(x-1)+7/2=-11/4 x+25/4.

Here's a graph of the situation just described:

enter image source here

Jul 16, 2015

Assuming that we want to find dy/dx:

Using implicit differentiation, we get: dy/dx = (3-4y)/(4x).

Solving for y first, we get: dy/dx = 11/(4x^2).

Explanation:

The question is posted under "Implicit Differentiation", so let's do it that way first:

4xy-3x-11=0

Leaving the function Implicit
In order to differentiate 4xy, we will need the product rule.

Remember that we are assuming that y is some function of x, so we have 4xy = 4xf(x) and we use the product rule to get:
the derivative is: 4f(x)+4xf'(x)

Back to this problem:

d/dx(4xy)-d/dx(3x)-d/dx(11)=d/dx(0)

4y+4xdy/dx-3=0

4xdy/dx = 3-4y

dy/dx = (3-4y)/(4x)

Making the function explicit

Solve 4xy-3x-11=0 for y

y = (3x+11)/(4x)

We could differentiate using the quotient rule, but it is perhaps simpler to rewrite again:

y = (3x)/(4x)+11/(4x)

= 3/4 +11/4x^-1

So
dy/dx = -11/4x^-2

= -11/(4x^2)

The answers are equivalent

To see that the answer are equivalent compare:

dy/dx = (3-4y)/(4x)

with

y = (3x+11)/(4x) and dy/dx = -11/(4x^2)

Using Implicit differentiation, there is still a y in the derivative. That is the price we pay for not making the function explicit before differentiating. If we substitute the solution for y, we get:

dy/dx = (3-4((3x+11)/(4x)))/(4x)

= (3-(3x+11)/x)/(4x)

= (3x-3x-11)/(4x^2)

= -11/(4x^2)