Implicit Differentiation

[pahuja]

I'm struggling w/ quotient rule in implicit differentiation questions specifically this:
Use process of implicit differentiation to find dy/dx given that 6x^2y^5 + (6y^5)/(7x^4) = 8xy

So far I got:

(12x)(y^5) + (6x^2)(5y^4)(dy/dx) + [(30y^4)(dy/dx)(7x^4)-(6y^5)(28x^3)] / (7x^4)^2 = (8)(y) + (8x)(1)(dy/dx)

Am stuck on what to do next, help is much appreciated!

 

[Dr.Peterson]

Good work so far.

The immediate next step is to simplify:

12xy^5 + 30x^2y^4(dy/dx) + [210x^4y^4(dy/dx) - 168x^3y^5] / (49x^8) = 8y + 8x(dy/dx)​

Now you just have to solve for dy/dx. So move all the terms containing that to the left side, and other terms to the right; then factor and divide.

12xy^5 + 30x^2y^4(dy/dx) + [210x^4y^4(dy/dx) - 168x^3y^5] / (49x^8) = 8y + 8x(dy/dx)​

(You might do other things like clearing fractions first.)

Can you show your attempt at the next step, so we can check it out?

 

[pahuja]

Good work so far.

The immediate next step is to simplify:

12xy^5 + 30x^2y^4(dy/dx) + [210x^4y^4(dy/dx) - 168x^3y^5] / (49x^8) = 8y + 8x(dy/dx)​

Now you just have to solve for dy/dx. So move all the terms containing that to the left side, and other terms to the right; then factor and divide.

12xy^5 + 30x^2y^4(dy/dx) + [210x^4y^4(dy/dx) - 168x^3y^5] / (49x^8) = 8y + 8x(dy/dx)​

(You might do other things like clearing fractions first.)

Can you show your attempt at the next step, so we can check it out?

So the following is what I did next, can you please ensure my final answer is correct and that I didn't miss anything:

Simplified: 12xy^5 + 30x^2y^4(dy/dx) + [210x^4y^4(dy/dx) - 168x^3y^5] / (49x^8) = 8y + 8x(dy/dx)

Multiplied every term by 49x^8: 588x^9y^5 + 1470x^10y^4(dy/dx)+210x^4y^4(dy/dx)-168x^3y^5 = 392x^8y + 392x^9(dy/dx)

Rearranged Terms: 1470x^10y^4(dy/dx)+210x^4y^4(dy/dx)-392x^9(dy/dx) = -588x^9y^5 +168x^3y^5+392x^8y

Factored out dy/dx: dy/dx(1470x^10y^4)+210x^4y^4-392x^9) = -588x^9y^5 +168x^3y^5+392x^8y

Final Answer: dy/dx = (-588x^9y^5 +168x^3y^5+392x^8y) / (1470x^10y^4)+210x^4y^4-392x^9)

 

[Dr.Peterson]

That's just what I got at that point.

You could, if you want, simplify a little more, by dividing numerator and denominator by 2x^3 (I think).

Edit: Looks like you can divide by 14x^3. See this answer: https://www.wolframalpha.com/input/?i=find+dy/dx+if+6+x^2+y^5+++(6+y^5)/(7+x^4)+=+8+x+y

 

[pahuja]

That's just what I got at that point.

You could, if you want, simplify a little more, by dividing numerator and denominator by 2x^3 (I think).

Edit: Looks like you can divide by 14x^3. See this answer: https://www.wolframalpha.com/input/?i=find+dy/dx+if+6+x^2+y^5+++(6+y^5)/(7+x^4)+=+8+x+y

Ok thankyou!