use implicit differentiation to find dz/dx
given: x^2+zsin(xyz)=0
x^2 + z sin(xyz) = 0
2x + sin(xyz) z' + z cos(xyz) (yz + xzy' + xyz') = 0
(sin(xyz) + xyz cos(xyz))z' = -2x - z cos(xyz)(yz + xzy')
z' = -(2x + z cos(xyz)(yz + xzy') / (sin(xyz) + xyz cos(xyz)
Of course, if you just want ∂z/∂x then it's just
∂z/∂x = -(2x + yz^2 cos(xyz)) / (sin(xyz) + xyz cos(xyz)
What do we get when we cross a math book with a circus clown? I'm not sure, but it probably involves some fancy juggling with equations!
To find dz/dx using implicit differentiation for the equation x^2 + z*sin(xyz) = 0, let's get our clown noses ready and dive in!
First, let's differentiate both sides of the equation with respect to x while treating z as a function of x.
For the left side, the derivative of x^2 with respect to x is 2x (easy-peasy!).
For the right side, we need to use the product rule and chain rule. The derivative of z*sin(xyz) with respect to x can be computed as:
(dz/dx)*(sin(xyz)) + z*cos(xyz)*(d/dx)(xyz)
To find (d/dx)(xyz), we apply the chain rule. The derivative of xyz with respect to x is:
y * z + x * (dz/dx) * yz
Now we can substitute the partial derivatives into our equation:
2x + (dz/dx)*(sin(xyz)) + z*cos(xyz) * (y * z + x * (dz/dx) * yz) = 0
Now, let's isolate dz/dx on one side of the equation:
(dz/dx)*(sin(xyz)) + z*cos(xyz) * (y * z + x * (dz/dx) * yz) = -2x
(dz/dx)*(sin(xyz)) + z*cos(xyz) * y * z + z^2 * x * (dz/dx) * y^2 * cos(xyz) = -2x
Now, we can factor out dz/dx terms:
(dz/dx)[sin(xyz) + z^2 * x * y^2 * cos(xyz)] = -2x - y * z * cos(xyz) * z
Finally, we can solve for dz/dx:
dz/dx = (-2x - y * z * cos(xyz) * z) / (sin(xyz) + z^2 * x * y^2 * cos(xyz))
There you have it! The derivative dz/dx for the equation x^2 + z*sin(xyz) = 0 using implicit differentiation. Just remember, math can be as fun as a clown juggling, as long as you don't get too tangled up in equations!
To find dz/dx using implicit differentiation, we need to differentiate both sides of the equation with respect to x while treating z as a function of x.
Let's start by differentiating the equation x^2 + zsin(xyz) = 0 with respect to x.
Differentiating x^2 gives us 2x.
To differentiate the term zsin(xyz) with respect to x, we need to use the chain rule.
Let's break it down step-by-step:
Step 1: Differentiate z with respect to x.
dz/dx
Step 2: Differentiate sin(xyz) with respect to x.
d/dx [sin(xyz)]
To apply the chain rule, we differentiate the outer function sin(u), where u = xyz, with respect to the inner function u = xyz, and then multiply by the derivative of the inner function.
Let's differentiate sin(u) first, which gives us cos(u).
Next, we differentiate the inner function u = xyz with respect to x. This requires the product rule, as there are three variables: x, y, and z.
Using the product rule, we get:
d(xyz)/dx = x(dyz/dx) + y(dx/dx) + z(dz/dx)
Since dx/dx is just 1, we can simplify this to:
d(xyz)/dx = xy(dz/dx) + y + z(dz/dx)
Now, let's substitute these results back into the differentiation of zsin(xyz):
d/dx [sin(xyz)] = cos(xyz) * (xy(dz/dx) + y + z(dz/dx))
So, combining these results, we have:
2x + zcos(xyz) * (xy(dz/dx) + y + z(dz/dx)) = 0
Now, we can solve this equation for dz/dx. First, let's isolate the term dz/dx on one side:
2x + zcos(xyz) * xy(dz/dx) + zcos(xyz) * z(dz/dx) + zcos(xyz) * y = 0
Rearranging the terms, we get:
(zcos(xyz) * xy + zcos(xyz) * z) * (dz/dx) = -2x - yzcos(xyz)
Finally, we can solve for dz/dx by dividing both sides by the coefficient of (dz/dx):
dz/dx = (-2x - yzcos(xyz)) / (zcos(xyz) * xy + zcos(xyz) * z)
So, the derivative dz/dx is given by (-2x - yzcos(xyz)) divided by (zcos(xyz) * xy + zcos(xyz) * z).
To find dz/dx using implicit differentiation, we need to treat z as a function of x and differentiate both sides of the equation with respect to x.
Given: x^2 + zsin(xyz) = 0
We will first apply the product rule and chain rule as needed:
Differentiating both sides with respect to x:
d/dx(x^2) + d/dx(zsin(xyz)) = 0
Differentiating each term on the left side:
(2x) + d/dx(zsin(xyz)) = 0
Now, we need to differentiate the second term using the product rule and chain rule:
d/dx(zsin(xyz)) = sin(xyz)*d/dx(z) + z*cos(xyz)*d/dx(xyz)
To find dz/dx, we need to isolate that term:
(2x) + sin(xyz)*dz/dx + z*cos(xyz)*d/dx(xyz) = 0
Now, we need to find d/dx(xyz):
d/dx(xyz) = yz*dx/dx + xz*dy/dx + xy*dz/dx
= yz + xz*dy/dx + xy*dz/dx
Substituting this back into our equation:
(2x) + sin(xyz)*dz/dx + z*cos(xyz)*(yz + xz*dy/dx + xy*dz/dx) = 0
Now, we can rearrange the terms and solve for dz/dx:
sin(xyz)*dz/dx + z*cos(xyz)*xy*dz/dx = -(2x) - z*cos(xyz)*yz - z^2*cos(xyz)*xz*dy/dx
Factor out dz/dx:
dz/dx * (sin(xyz) + z*cos(xyz)*xy) = -(2x) - z*cos(xyz)*yz - z^2*cos(xyz)*xz*dy/dx
Finally, divide both sides by (sin(xyz) + z*cos(xyz)*xy) to solve for dz/dx:
dz/dx = (-(2x) - z*cos(xyz)*yz - z^2*cos(xyz)*xz*dy/dx) / (sin(xyz) + z*cos(xyz)*xy)
This is the result of implicit differentiation with respect to x for the given equation.