The differential of the function y=(x^2+6)^3 is
dy=______dx.
When x=2 and dx=0.05, the differential dy=_______?
A) dy=______dx?
B) the differential dy=_______?
dy/dx = 3(x^2 + 6)^2
dy = 3(x^2 + 6)^2 dx
when x = 2 and dx = .05
dy = ......
just plug in the above values.
To find the differential dy=______dx of the function y=(x^2+6)^3, we need to take the derivative of the function with respect to x.
First, let's find the derivative of y with respect to x. We can use the chain rule to differentiate this composite function.
First, consider the inner function u(x) = x^2 + 6. The derivative of u with respect to x, du/dx, is given by 2x.
Next, consider the outer function v(u) = u^3. The derivative of v with respect to u, dv/du, is given by 3u^2.
Now, applying the chain rule, we have: dy/dx = (dv/du) * (du/dx)
= 3u^2 * 2x
= 6x(x^2 + 6)^2
So, the derivative of y with respect to x is dy/dx = 6x(x^2 + 6)^2.
Now, to find dy, we can substitute the given values x=2 and dx=0.05 into the derivative expression.
dy = 6x(x^2 + 6)^2 * dx
= 6(2)(2^2 + 6)^2 * 0.05
= 6(2)(8 + 6)^2 * 0.05
= 6(2)(14)^2 * 0.05
= 6(2)(196) * 0.05
= 235.2 * 0.05
= 11.76
Therefore, when x=2 and dx=0.05, the differential dy is 11.76.
So, the answers are:
A) dy=11.76*dx
B) The differential dy=11.76.