Find dy/dx.
y^3 = 8x^3 + 4x
3y^2 dy=24x^2 dx + 4dx
dy/dx=(8x^2+1)/(y^2) check that
bob probably meant to type
dy/dx = (8x^2 + 4/3)/(y^2)
To find dy/dx, we need to differentiate both sides of the equation with respect to x using the power rule of differentiation. The power rule states that if we have a function of the form f(x) = x^n, then the derivative of f(x) with respect to x is given by d/dx(f(x)) = n*x^(n-1).
Differentiating both sides of the equation:
d/dx(y^3) = d/dx(8x^3 + 4x)
Applying the power rule to the left-hand side, we get:
3y^2*(dy/dx) = d/dx(8x^3 + 4x)
Now we can differentiate the right-hand side term by term:
3y^2*(dy/dx) = d/dx(8x^3) + d/dx(4x)
The derivative of 8x^3 with respect to x is 24x^2, and the derivative of 4x with respect to x is 4.
Substituting these values back into the equation, we have:
3y^2*(dy/dx) = 24x^2 + 4
To isolate dy/dx, we divide both sides of the equation by 3y^2:
dy/dx = (24x^2 + 4) / (3y^2)
Therefore, the derivative of y with respect to x, dy/dx, is equal to (24x^2 + 4) / (3y^2).