Find the second derivative of x(x+1)^3
have you found the first derivative?
yes, I think I found the second too. 6(x+1)(2x+1)?
x(x+1)^3
1st derivative
(x)*3(x + 1)^2*(1) + (x + 1)^3 * (1)
3x(x + 1)^2 + (x + 1)^3
Is this what you have for the 1st derivative?
I'm not a tutor, just a helper, but I've checked my answer twice.
Allison , you are right,
it is 6(x+1)(2x+1) after factoring and simplifying.
Helper is correct too.
To find the second derivative of the expression x(x+1)^3, we need to apply the chain rule and the product rule.
Step 1: Use the product rule to find the first derivative.
Let's consider the expression as two factors: x and (x+1)^3.
The derivative of x is simply 1.
To find the derivative of (x+1)^3, we use the chain rule. Let u = x+1, so (x+1)^3 = u^3.
The derivative of u^3 is 3u^2 * du/dx.
The derivative of u with respect to x is simply 1, as it's a linear function. So, du/dx = 1.
Therefore, the first derivative of x(x+1)^3 is:
1 * (x+1)^3 + x * 3(x+1)^2 * 1
Simplifying this expression gives:
(x+1)^3 + 3x(x+1)^2
Step 2: Use the product rule again to find the second derivative.
Let's differentiate the expression (x+1)^3 + 3x(x+1)^2 using the product rule.
The first term (x+1)^3 has a constant multiplier of 1, so its derivative will be 3(x+1)^2.
For the second term 3x(x+1)^2, we differentiate each part separately. The derivative of 3x with respect to x is simply 3.
The derivative of (x+1)^2 using the chain rule is 2(x+1) * du/dx.
Let's consider u = x+1, so (x+1)^2 = u^2.
The derivative of u^2 with respect to x is 2u * du/dx, where du/dx = 1.
Simplifying this expression gives 2(x+1).
Now we can apply the product rule for the second term: 3x * 2(x+1) + 3 * (x+1)^2.
Simplifying this expression gives:
6x(x+1) + 3(x+1)^2
Therefore, the second derivative of x(x+1)^3 is:
3(x+1)^2 + 6x(x+1) + 3(x+1)^2
Simplifying further gives:
6x(x+1) + 6(x+1)^2
which can be written as:
6(x(x+1) + (x+1)^2)
Finally, we can factor out a common term to obtain the simplified second derivative:
6(x+1)[x + (x+1)]