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.