Find the derivative, implicit differentiation (no calculator)

Question

Find dy/dx at the point (4,1) when 3xy²+4lny = x²-4y².

Answer choices:

A. 11/36

B. 5/36

C. 11/20

D. 5/20

Answer 1

I'm going to put the exponents back into your equation:

3xy² + 4 ln y = x² - 4y²

In this "implicit differentiation" problem, you take the derivative (with respect to x) of both sides of the equation:

d/dx(3xy² + 4 ln y) = d/dx(x² - 4y²)

How to do this? We have two terms on the left side of the equation, two terms on the right.

Let's apply d/dx to each of these four terms individually. (The incantation you probably memorized is "The derivative of the sum is the sum of the derivatives".)

d/dx(3xy²) + d/dx(4 ln y) = d/dx(x²) - d/dx(4y²)

Okay, now let's take these four derivatives (with respect to x) individually:

In other words, let's take these four d/dx's individually.

The easiest one is number three: d/dx(x²) = 2x

Let's bite the bullet and do the other three terms.

The first one is

d/dx(3xy²) = 3 d/dx(xy²) (move the factor 3 to the front)

= 3(y² + 2xy (dy/dx)) (formula for the derivative of the product. In this case, the product is

x times y²)

The second one is

d/dx(4 ln y) = 4 d/dx(ln y) (move the factor 4 to the front)

= (4/y)(dy/dx) (chain rule, and formula for derivative of ln)

The fourth one is

d/dx(4y²) = 4 d/dx(y²) (move the factor 4 to the front)

= 4 times 2 y (dy/dx) (chain rule)

Putting the four terms together,

3(y² + 2xy (dy/dx)) + (4/y)(dy/dx) = 2x - 8 y (dy/dx)

Distributing the 3,

3y² + 6xy (dy/dx)) + (4/y)(dy/dx) = 2x - 8 y (dy/dx)

Substituting x = 4 and y = 1,

3 + 24(dy/dx) + 4(dy/dx) = 8 - 8(dy/dx)

Subtract 3 from each side, to get rid of the 3:

24(dy/dx) + 4(dy/dx) = 5 - 8(dy/dx)

Move the 8(dy/dx) to the left side, so that all the dy/dx are on the same side:

24(dy/dx) + 4(dy/dx)+ 8(dy/dx) = 5

Now 24 plus 4 plus 8 is 36:

36(dy/dx) = 5

Divide both sides by 36:

dy/dx = 5/36, which is answer choice B. Thank you for sticking with this till the end.