Implicit Differentiation Clarification

[Dick]

Looks ok to me.

 

[Asphyxiated]

$$y = sin(xy)$$

$$\frac {dy}{dx} = cos(xy) * (x\frac{dy}{dx} +y)$$

$$\frac {dy}{dx} = (x\frac{dy}{dx})cos(xy)+(y)cos(xy)$$

$$\frac {dy}{dx}(1 -(x)cos(xy))= (y)cos(xy)$$

$$\frac {dy}{dx}= \frac{y*cos(xy)} {1 -x*cos(xy)}$$

 

[Asphyxiated]

And another:

$$x = cos^{2} (y)$$

$$1 = (2cos(y))(-sin(y))(\frac{dy}{dx})$$

$$\frac {1} {(2cos(y))(-sin(y))} = \frac {dy}{dx}$$

 

[Asphyxiated]

Hopefully getting the hang of this:

$$y = e^{x+2y}$$

$$\frac {dy}{dx} = e^{x+2y} * (1+2\frac{dy}{dx})$$

$$\frac {dy}{dx} = e^{x+2y} + 2\frac{dy}{dx}e^{x+2y}$$

$$\frac {dy}{dx} - 2\frac{dy}{dx}e^{x+2y} = e^{x+2y}$$

$$\frac {dy}{dx}(1 - 2e^{x+2y}) = e^{x+2y}$$

$$\frac {dy}{dx} = \frac {e^{x+2y}} {(1 - 2e^{x+2y})}$$

 

[Dick]

Yes, I think you've gotten the hang of it.

 

[Asphyxiated]

One more for good measure? It's the last question in the section any way

$$y^{2} = ln (2x+3y)$$

$$2y\frac{dy}{dx} = \frac {1} {2x+3y} * (2+3\frac{dy}{dx})$$

$$2y\frac{dy}{dx} = \frac {2} {2x+3y} + \frac {3}{2x+3y}\frac{dy}{dx}$$

$$2y\frac{dy}{dx} - \frac {3}{2x+3y}\frac{dy}{dx} = \frac {2} {2x+3y}$$

$$\frac{dy}{dx}(2y - \frac {3}{2x+3y})= \frac {2} {2x+3y}$$

which is to say:

$$\frac{dy}{dx}(\frac {(2y)(2x+3y)}{2x+3y} - \frac {3}{2x+3y})= \frac {2} {2x+3y}$$

$$\frac{dy}{dx}(\frac {(2y)(2x+3y)-3}{2x+3y})= \frac {2} {2x+3y}$$

$$\frac{dy}{dx}= \frac {\frac {2} {2x+3y}} {\frac {(2y)(2x+3y)-3}{2x+3y}}$$

$$\frac{dy}{dx}= \frac {2} {2x+3y} *\frac {2x+3y}{(2y)(2x+3y)-3}$$

$$\frac{dy}{dx}= \frac {(2)(2x+3y)} {(2x+3y)(2y)(2x+3y)-3}$$

$$\frac{dy}{dx}= \frac{2} {(2y)(2x+3y)-3}$$

 

[Dick]

Last one?? Promise? That one looks good to me.

 

[Asphyxiated]

Yep that's the last one, thanks 1000x for all the help and patience Dick!

 

[Dick]

Very welcome. It's nice to see someone learn from mistakes and learn not to repeat them. Wish that happened more often around here. You are welcome back anytime!

 

[Asphyxiated]

Oh I am sure that I will have more questions soon enough as I am only on chapter 3 of 13 in this particular book, but I am glad to hear that I didn't ask too many questions as I had suspected.