find dy/dx
y=3u^2-6u+2,u=v^2-1;v=2x
dy/du = 6u - 6
du/dv = 2v
dv/dx = 2
dy/dx = (dy/du)(du/dv)(dv/dx)
= (6u-6)(2v)(2)
= 24v(u-1)
If necessary, use substitution of the originals to express in terms of whichever variable you want
To find dy/dx, we need to use the chain rule since y depends on u, and u depends on v, and v depends on x.
Let's start by finding du/dv:
Given u = v^2 - 1
To find du/dv, we take the derivative of u with respect to v.
Differentiating both sides of the equation with respect to v:
du/dv = d/dv (v^2 - 1)
= 2v
Now let's find dv/dx:
Given v = 2x
To find dv/dx, we take the derivative of v with respect to x.
Differentiating both sides of the equation with respect to x:
dv/dx = d/dx (2x)
= 2
Now we can find dy/dx using the chain rule:
dy/dx = dy/du * du/dv * dv/dx
Substituting the derivatives we found earlier, we have:
dy/dx = (d/dv (3u^2 - 6u + 2)) * (2v) * 2
= (6u - 6) * (2v) * 2
= 12v(3u - 3)
= 12(2x)(3(v^2 - 1) - 3)
Simplifying further:
dy/dx = 24x(3v^2 - 3 - 3)
= 24x(3v^2 - 6)
Finally, substituting v = 2x:
dy/dx = 24x(3(2x)^2 - 6)
= 24x(12x^2 - 6)
= 288x^3 - 144x
To find dy/dx, we can use the chain rule of differentiation. The chain rule states that if we have a composite function, we can differentiate it by differentiating the outer function with respect to the inner function, and then multiplying it by the derivative of the inner function with respect to the independent variable.
In this case, we have y = 3u^2 - 6u + 2, where u = v^2 - 1, and v = 2x. Let's start by finding du/dv and dv/dx.
Given u = v^2 - 1, we can find du/dv by differentiating u with respect to v:
du/dv = 2v.
Given v = 2x, we can find dv/dx by differentiating v with respect to x:
dv/dx = 2.
Now, we can apply the chain rule. We differentiate the outer function y = 3u^2 - 6u + 2 with respect to the inner function u, and then multiply it by the derivative of u with respect to x.
dy/dx = (dy/du) * (du/dv) * (dv/dx).
First, let's find dy/du by differentiating y with respect to u:
dy/du = d/dx (3u^2 - 6u + 2) = 6u - 6.
Now, we can substitute the values we calculated earlier:
dy/dx = (6u - 6) * (2v) * (2).
But we need to express u and v in terms of x:
u = v^2 - 1 = (2x)^2 - 1 = 4x^2 - 1.
Now we can substitute u = 4x^2 - 1 and dv/dx = 2 into the equation:
dy/dx = (6(4x^2 - 1) - 6) * (2(2x)) * (2).
Simplifying the expression:
dy/dx = (24x^2 - 6) * (4x) * (2)
= (192x^3 - 48x).
Therefore, the derivative dy/dx of the function y = 3u^2 - 6u + 2, where u = v^2 - 1 and v = 2x, is dy/dx = 192x^3 - 48x.