Question

If 6x^2+xy-cos(y)=10, what is dy/dx?

Answer 1

EJ: As you'll see, this equation has NOT been solved for y, and it would be quite difficult to solve it for y.
therefore, to find the derivative dy/dx, use IMPLICIT DIFFERENTIATION.

Answer 2

Enclose your original equation in parentheses and apply the derivative operator (d/dx) to the whole thing.
Differentiate term by term.
Note that xy is a product, which means you MUST apply the product rule, while remembering that y is presumed to be a function of x.

Answer 3

EJ?

Answer 4

Thank you! Can you explain getting the derivative of the function? There are a lot of terms.

Answer 5

I can guide you thru the process. First, what is the derivative (with respect to x) of 6x^2?

Answer 6

12x

Answer 7

Right. Given that xy is a product, and that y is assumed to be a function of x, what is the derivative of xy? ( )( )+( )( )

Answer 8

(x)(dy/dx)+(y)(dx/dx)

Answer 9

Yes, very good. And dx/dx=?

Answer 10

what about -cos(y)?

Answer 11

dx/dx=1

Answer 12

First: dx/dx= 1. so your derivative of xy is therefore x(dy/dx)+y. OK?

Answer 13

yes

Answer 14

What is the derivative of -cos y?
Recall that y is assumed to be a function of x, so we must use the Chain Rule and differentiate y.
What do you recall about the derivative of cos x?
(d/dx)(cos x) = ?

Answer 15

-sinx

Answer 16

Right, and so the derivative of -cos x is what?

Answer 17

sinx

Answer 18

Right. Now, knowing that y is supposedly a function of x,
what is the derivative with respect to x of -cos y?

Answer 19

sinx (dy/dx)

Answer 20

That's great! right on target!

Answer 21

What's the derivative of 10?

Answer 22

perfect thank you! Then 10 goes to zero and i solve for dy/dx right?

Answer 23

That's right. Confident that you can finish up?

Answer 24

yes thank you so much

Answer 25

My great pleasure, EJ! "See" you again on Open Study.