# How do you differentiate #x^3 yz^2+w^2 x^2 y-x^3+wsiny=44#?

##### 1 Answer

#### Explanation:

I'll assume that

You can calculate the *partial derivative* of *constants*.

When you differentiate with respect to

On the other hand,

So, let's start with the partial derivative of

Now, you can calculate

#(del)/(delx)(yx^2w^2) = y * [d/dx(x^2) * w^2 + x^2 * (del)/(delx)(w^2)]#

#(del)/(delx)(yx^2w^2) = y * (2x * w^2 + x^2 * 2w * (delw)/(delx))#

#(del)/(delx)(yx^2w^2) = 2xyw^2 + 2x^2yw(delw)/(delx)#

Plug this back into the calculation with respect to

#yz^2 * 3x^2 + 2xyw^2 + 2x^2yw(delw)/(delx) - 3x^2 + siny * (delw)/(delx) = 0#

Rearrange this to isolate

#(delw)/(delx)(2x^2yw + siny) = +3x^2 - 2xyw^2 - 3x^2yz^2#

#(delw)/(delx) = color(green)((3x^2(1 - yz^2) - 2xyw^2)/(2x^2yw + siny))#

Next, take the partial derivative of

#(del)/(dely)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/(dy)(44)#

This will get you

#x^3z^2 + x^2 * [(del)/(dely)(w^2) * y + w^2 * d/(dy)(y)] - 0 + [(del)/(dely)(w) * siny + w * d/(dy)(siny)] = 0#

#x^3z^2 + x^2 * (2w * (delw)/(dely) + w^2) + (delw)/(dely) * siny + w * cosy = 0#

#x^3z^2 + 2x^2w(delw)/(dely) + x^2w^2 + (delw)/(dely) * siny + wcosy = 0#

Once again, rearrange to get

#(delw)/(dely)(2x^2w + siny) = -x^3z^2 - x^2w^2 - wcosy#

#(delw)/(dely) = color(green)(-(x^2(xz^2 + w^2) + wcosy)/(2x^2 + siny))#

Finally, take the partial derivative of

#(del)/(delz)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/(dz)(44)#

#2x^3yz + 2x^2yw(delw)/(delz) - 0 + siny * (delw)/(delz) = 0#

Rearrange to get

#(delw)/(delz)(2x^2yw + siny) = -2x^3yz#

#(delw)/(delz) = color(green)(-(2x^3yz)/(2x^2yw + siny))#