Use the Chain Rule, in Leibniz notation, to find dy/dx at the indicated value of x. y=u(u^2+3)^3, u=(x+3)^2, x=-2
AAAaannndd the bot gets it wrong yet again!
it got this far ok
dy/dx = (dy/du) * (du/dx) = 6u^2(u^2+3)^2 * 2(x+3)
But blew it when using x = -2, u = (-2+3)^2 = 1
dy/dx at x = -2 is thus
6*1(1+3)^2 * 2(-2+3) = 6 * 4^2 * 2 = 192
how it could work with all those positive values and end up with a negative answer is beyond me.
We can begin by taking the derivative of y with respect to u using the power rule and the chain rule:
dy/du = 3u(u^2+3)^2 * 2u
After simplifying, we get:
dy/du = 6u^2(u^2+3)^2
Next, we can substitute u = (x+3)^2 into the expression for dy/du:
dy/dx = (dy/du) * (du/dx) = 6u^2(u^2+3)^2 * 2(x+3)
Now we can substitute x = -2 and simplify:
u = (-2+3)^2 = 1
dy/dx = 6(1)^2(1^2+3)^2 * 2(-2+3) = -288
Therefore, dy/dx = -288 when x = -2.