Question

You are given the following equation. 16x2 + 25y2 = 400 (a) Find dy / dx by implicit differentiation. dy / dx = Correct: Your answer is correct. (b) Solve the equation explicitly for y and differentiate to get dy / dx in terms of x. (Consider only the first and second quadrants for this part.) dy / dx = Incorrect: Your answer is incorrect. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). (Do this on paper. Your teacher may ask you to turn in this work.)

Answer 1

a)
`(dy)/(dx)=-(16x)/(25y)`

b)
`(dy)/(dx)=-\frac{4x}{25\sqrt{1-(x^2)/(25)}}`

c) The two expressions match

Answer:

a)

The equation in this problem is

`16x^2+25y^2=400`

Here, we want to find
`(dy)/(dx)` by implicit differentiation.

To do that, we apply the operator
`(d)/(dx)` on each term of the equation. We have:

`(d)/(dx)(16 x^2)=32x`

`(d)/(dx)(25y^2)=50y (dy)/(dx)` (by applying composite function rule)

`(d)/(dx)(400)=0`

Therefore, the equation becomes:

`32x+50y(dy)/(dx)=0`

And re-arranging for dy/dx, we get:

`50(dy)/(dx)=-32x\\(dy)/(dx)=-(32x)/(50y)=-(16x)/(25y)`

b)

Now we want to solve the equation explicitly for y and then differentiate to find dy/dx. The equation is:

`16x^2+25y^2=400`

First, we isolate y, and we find:

`25y^2=400-16x^2\\y^2=16-(16)/(25)x^2`

And taking the square root,

`y=\pm \sqrt{16-(16)/(25)x^2}=\pm 4\sqrt{1-(x^2)/(25)}`

Here we are told to consider only the first and second quadrants, so those where y > 0; so we only take the positive root:

`y=4\sqrt{1-(x^2)/(25)}`

Now we differentiate this function to find dy/dx; using the chain rule, we get:

`(dy)/(dx)=4[(1)/(2)(1-(x^2)/(25))^{-(1)/(2)}\cdot(-(2x)/(25))]=-\frac{4x}{25\sqrt{1-(x^2)/(25)}}` (2)

c)

Now we want to check if the two solutions are consistent.

To do that, we substitute the expression that we found for y in part b:

`y=4\sqrt{1-(x^2)/(25)}`

Into the solution found in part a:

`(dy)/(dx)=-(16x)/(25y)`

Doing so, we find:

`(dy)/(dx)=-\frac{16x}{25(4\sqrt{1-(x^2)/(25)})}=-\frac{4x}{25\sqrt{1-(x^2)/(25)}}` (1)

We observe that expression (1) matches with expression (2) found in part b: therefore, we can conclude that the two solutions are coeherent with each other.