If Y=(x^3+1)6, find d^2y/dx^2

y = (x^3 + 1)*6,

Differentiate it twice.
dy/dx = 6*(3x^2) = 18x^2
d^2y/dx^2 = 36 x

Have you learned the rule for differentiating a*x^n ?
d/dx (a*x^n) = n*a*x^(n-1)
That is all you needed to know for that problem.

Did you mean y = (x^3 + 1)^6 ?

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To find the second derivative of Y with respect to x, d^2y/dx^2, we will need to differentiate the function Y twice.

First, let's find the first derivative of Y with respect to x, dy/dx.

Given Y = (x^3 + 1)6, we can use the power rule to differentiate each term with respect to x. The power rule states that for a term of the form ax^n, the derivative with respect to x is nax^(n-1).

So, applying the power rule to the term (x^3 + 1), we get:

dy/dx = 6 * d/dx (x^3 + 1)

Now, differentiating each term separately using the power rule:

dy/dx = 6 * (3x^2 + 0)

Simplifying further:

dy/dx = 18x^2

Now, let's find the second derivative, d^2y/dx^2.

We will differentiate dy/dx with respect to x using the power rule:

d^2y/dx^2 = d/dx (18x^2)

Again, applying the power rule to the term 18x^2:

d^2y/dx^2 = 2 * 18x^(2-1)

Simplifying further:

d^2y/dx^2 = 36x

Therefore, the second derivative of Y with respect to x, d^2y/dx^2, is equal to 36x.