find the nth derivative of 2/(x+1)
what i have so far is (?)(x+1)^(-n-1)
i having trouble with finding the first number without using factorial. 2^(n-1) works for the first few but not after that
To find the nth derivative of 2/(x+1), we can use the power rule for derivatives.
First, let's rewrite the function using negative exponents: 2(x+1)^(-1).
Now, let's start by taking the first derivative. Using the power rule, the derivative of (x+1)^(-1) is -1(x+1)^(-2). Notice that the power decreases by 1, and we multiply by the original power.
To find the second derivative, we differentiate again using the power rule. The derivative of -1(x+1)^(-2) is 2(x+1)^(-3). Again, the power decreases by 1, and we multiply by the original power.
Now, let's observe the pattern:
- The first derivative had a power of -2.
- The second derivative had a power of -3.
- The third derivative should have a power of -4.
So, we can generalize that the nth derivative will have a power of -n-1.
To determine the coefficient in front of (x+1)^(-n-1), we need to determine a pattern or a formula. We notice that the coefficients seem to be alternating between positive and negative. Specifically, the sign of the coefficient depends on whether n is even or odd.
If n is even, the coefficient will be positive: (+1)^n.
If n is odd, the coefficient will be negative: (-1)^n.
Combining these observations, we can find the nth derivative as follows:
- Determine the coefficient: 2 * (-1)^n or 2 * (+1)^n, depending on whether n is odd or even.
- Determine the new power of (x+1): -n-1.
Therefore, the nth derivative of 2/(x+1) is given by the formula:
2 * (-1)^n * (x+1)^(-n-1).
This formula will give you the correct coefficients and powers for any value of n.