Use implicit differentiation to find dz/dx if π^z = π₯π¦π§
To find dz/dx using implicit differentiation, we will differentiate both sides of the equation π^z = π₯π¦π§ with respect to x.
Step 1: Differentiate both sides of the equation with respect to x, keeping in mind that z is a function of x and y.
On the left side, we differentiate π^z with respect to x using the chain rule, which states that d(u^v)/dx = v(u^(v-1)) * du/dx + u^v * ln(u) * dv/dx.
So, differentiating π^z with respect to x, we get: d(π^z)/dx = π^z * dz/dx.
On the right side, we have three terms: π₯, π¦, and π§. Since we're differentiating with respect to x, π¦ and π§ are considered constants.
Differentiating π₯ with respect to x gives us 1.
Differentiating π¦ with respect to x gives us 0 since π¦ is a constant.
Differentiating π§ with respect to x gives us dz/dx.
Therefore, the right side differentiates to 1 * π¦ * π§ + π₯ * 0 + π₯ * π¦ = π¦π§.
Step 2: Equate the left and right side derivatives and solve for dz/dx.
π^z * dz/dx = π¦π§.
Divide both sides by π^z:
dz/dx = π¦π§ / π^z.
So, dz/dx = π¦π§ / π^z.
To find the derivative dz/dx using implicit differentiation, we'll differentiate both sides of the equation with respect to x while treating y and z as functions of x.
Let's start by differentiating the left-hand side (LHS). Since e^z is a composition of functions, we'll use the chain rule. Remember that the derivative of e^u with respect to u is e^u times the derivative of u with respect to x.
d/dx (e^z) = e^z * dz/dx
Next, we'll differentiate the right-hand side (RHS). In this case, all three variables x, y, and z are multiplied together, so we'll need to use the product rule. The product rule states that if we have two functions u and v that are differentiable with respect to x, then the derivative of their product is given by:
d/dx (u * v) = u * dv/dx + v * du/dx
For π₯π¦π§, let's treat y and z as functions of x, so we'll have:
d/dx (π₯π¦π§) = x * d/dx(yz) + y * d/dx(xz) + z * d/dx(xy)
To simplify this expression, we'll now differentiate each term one by one:
d/dx(yz) = y * dz/dx + z * dy/dx
d/dx(xz) = x * dz/dx + z * dx/dx (dx/dx = 1)
d/dx(xy) = x * dy/dx + y * dx/dx (dx/dx = 1)
Now, substituting these derivatives back into our result above, we get:
x * d/dx(yz) + y * d/dx(xz) + z * d/dx(xy) = x * (y * dz/dx + z * dy/dx) + y * (x *dz/dx + z) + z * (x * dy/dx + y)
Simplifying further, we have:
x * (y * dz/dx + z * dy/dx) + y * (x * dz/dx + z) + z * (x * dy/dx + y) = x * y * dz/dx + x * z * dy/dx + y * x * dz/dx + y * z + z * x * dy/dx + z * y
Combining like terms, we get:
x * y * dz/dx + x * z * dy/dx + y * x * dz/dx + y * z + z * x * dy/dx + z * y = x * y * dz/dx + y * z + z * x * dy/dx + z * y
Now, equating the derivatives from the LHS and RHS, we have:
e^z * dz/dx = x * y * dz/dx + y * z + z * x * dy/dx + z * y
Next, rearrange the equation to isolate dz/dx:
e^z * dz/dx - x * y * dz/dx = y * z + z * x * dy/dx + z * y
(dz/dx) * (e^z - x * y) = z * y + z * x * dy/dx
Finally, divide both sides by (e^z - x * y) to solve for dz/dx:
dz/dx = (z * y + z * x * dy/dx) / (e^z - x * y)
And there you have it! The derivative dz/dx using implicit differentiation is given by (z * y + z * x * dy/dx) / (e^z - x * y).