How do you differentiate x^3 yz^2+w^2 x^2 y-x^3+wsiny=44x3yz2+w2x2yx3+wsiny=44?

1 Answer
Aug 26, 2015

(delw)/(delx) = (3x^2(1 - yz^2) - 2xyw^2)/(2x^2yw + siny)wx=3x2(1yz2)2xyw22x2yw+siny
(delw)/(dely) = -(x^2(xz^2 + w^2) + wcosy)/(2x^2 + siny)wy=x2(xz2+w2)+wcosy2x2+siny
(delw)/(delz) = -(2x^3yz)/(2x^2yw + siny)wz=2x3yz2x2yw+siny

Explanation:

I'll assume that ww is your dependent variable and that xx, yy, and zz are all independent variables.

You can calculate the partial derivative of ww with respect to each independent variable by using implicit differentiation. Keep in mind, when you differentiate with respect to xx, yy and zz are treated like constants.

When you differentiate with respect to yy, xx and zz are treated like constants. And so on.

On the other hand, ww is treated like a function that depends on all three independent variables. This means that you're going to use the chain rule whenever you derive ww with respect to an independent variable.

So, let's start with the partial derivative of ww with respect to xx. You're dealing with an equation, which means that you need to differentiate both sides with respect to xx.

(del)/(delx)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/dx(44)x(x3yz2+w2x2yx3+wsiny)=ddx(44)

Now, you can calculate (del)/(delx)(yx^2w^2)x(yx2w2) by using the product rule. Remember that yy is treated like a constant

(del)/(delx)(yx^2w^2) = y * [d/dx(x^2) * w^2 + x^2 * (del)/(delx)(w^2)]x(yx2w2)=y[ddx(x2)w2+x2x(w2)]

(del)/(delx)(yx^2w^2) = y * (2x * w^2 + x^2 * 2w * (delw)/(delx))x(yx2w2)=y(2xw2+x22wwx)

(del)/(delx)(yx^2w^2) = 2xyw^2 + 2x^2yw(delw)/(delx)x(yx2w2)=2xyw2+2x2ywwx

Plug this back into the calculation with respect to xx to get

yz^2 * 3x^2 + 2xyw^2 + 2x^2yw(delw)/(delx) - 3x^2 + siny * (delw)/(delx) = 0yz23x2+2xyw2+2x2ywwx3x2+sinywx=0

Rearrange this to isolate (delw)/(delx)wx on one side

(delw)/(delx)(2x^2yw + siny) = +3x^2 - 2xyw^2 - 3x^2yz^2wx(2x2yw+siny)=+3x22xyw23x2yz2

(delw)/(delx) = color(green)((3x^2(1 - yz^2) - 2xyw^2)/(2x^2yw + siny))wx=3x2(1yz2)2xyw22x2yw+siny

Next, take the partial derivative of ww with respect to yy.

(del)/(dely)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/(dy)(44)y(x3yz2+w2x2yx3+wsiny)=ddy(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)] = 0x3z2+x2[y(w2)y+w2ddy(y)]0+[y(w)siny+wddy(siny)]=0

x^3z^2 + x^2 * (2w * (delw)/(dely) + w^2) + (delw)/(dely) * siny + w * cosy = 0x3z2+x2(2wwy+w2)+wysiny+wcosy=0

x^3z^2 + 2x^2w(delw)/(dely) + x^2w^2 + (delw)/(dely) * siny + wcosy = 0x3z2+2x2wwy+x2w2+wysiny+wcosy=0

Once again, rearrange to get (delw)/(dely)wy isoltated on one side

(delw)/(dely)(2x^2w + siny) = -x^3z^2 - x^2w^2 - wcosywy(2x2w+siny)=x3z2x2w2wcosy

(delw)/(dely) = color(green)(-(x^2(xz^2 + w^2) + wcosy)/(2x^2 + siny))wy=x2(xz2+w2)+wcosy2x2+siny

Finally, take the partial derivative of ww with respect to zz. This one will be easier to calculate because you won't use the product rule

(del)/(delz)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/(dz)(44)z(x3yz2+w2x2yx3+wsiny)=ddz(44)

2x^3yz + 2x^2yw(delw)/(delz) - 0 + siny * (delw)/(delz) = 02x3yz+2x2ywwz0+sinywz=0

Rearrange to get (delw)/(delz)wz alone on one side

(delw)/(delz)(2x^2yw + siny) = -2x^3yzwz(2x2yw+siny)=2x3yz

(delw)/(delz) = color(green)(-(2x^3yz)/(2x^2yw + siny))wz=2x3yz2x2yw+siny