Find the derivatives given Z= 3y² where y = 2x + 5
To find the derivative of Z with respect to x, we need to use the chain rule.
Given:
Z = 3y^2
y = 2x + 5
First, we need to find the derivative of y with respect to x:
dy/dx = d/dx(2x + 5)
dy/dx = 2
Now, we can find the derivative of Z with respect to x using the chain rule:
dZ/dx = d(Z)/dy * dy/dx
dZ/dx = 6y * 2 (using the power rule for differentiation)
dZ/dx = 12y
Substitute y = 2x + 5 back into the equation:
dZ/dx = 12(2x + 5)
dZ/dx = 24x + 60
Therefore, the derivative of Z = 3y^2, where y = 2x + 5, with respect to x is 24x + 60.