find the nth derivative of 2/(x+1)
f(x)=2/(x+1)
df/dx = -2/(x+1)^2
d²f/dx² = 4/(x+1)³
d³f/dx² = -12/(x+1)^4
...
d10f/dx10 = 7257600/(x+1)^11
The last derivative is intended to be a test for your "rule", which your teacher would like you to discover.
If you wish, you could send in your nth derivative for a check.
To find the nth derivative of the function f(x) = 2/(x+1), we will use the power rule for derivatives.
Step 1: Write the function in the form f(x) = 2(x+1)^(-1).
Step 2: Find the first derivative by applying the power rule. The power rule states that d/dx[x^n] = n * x^(n-1).
f'(x) = -2(x+1)^(-2).
Step 3: To find the second derivative, differentiate f'(x) using the power rule again.
f''(x) = 2 * (-2)(x+1)^(-3) = 4(x+1)^(-3).
Step 4: Continue differentiating for each subsequent derivative.
f'''(x) = 4 * (-3)(x+1)^(-4) = -12(x+1)^(-4).
f''''(x) = -12 * (-4)(x+1)^(-5) = 48(x+1)^(-5).
...
f^(n)(x) = (-1)^n * n! * 2 * (x+1)^(-n-1).
Therefore, the nth derivative of f(x) = 2/(x+1) is f^(n)(x) = (-1)^n * n! * 2 * (x+1)^(-n-1).
To find the nth derivative of a function, we can use the power rule for differentiation and the quotient rule repeatedly. Let's proceed step by step to find the nth derivative of the function 2/(x + 1):
Step 1: Determine the first derivative.
Using the quotient rule, the first derivative of the function is given by:
f'(x) = [2 * (x + 1)^0 - 0 * 2 * (x + 1)^(-1)] / (x + 1)^2
= 2 / (x + 1)^2
Step 2: Determine the second derivative.
To find the second derivative, we differentiate f'(x) using the quotient rule again:
f''(x) = [0 * (x + 1)^2 - 2 * 2 * (x + 1)^1] / (x + 1)^4
= -4 / (x + 1)^3
Step 3: Proceed with differentiating higher derivatives.
We can see that a pattern is emerging. Each time we differentiate, we end up with a factor of (-1)^n multiplied by (n!)/(x + 1)^(n+1), where n is the number of times the function has been differentiated.
Based on this pattern, we can conclude that the nth derivative of 2/(x + 1) is:
f^(n) (x) = (-1)^n * (n!)/(x + 1)^(n+1),
where n is a positive integer.
So, to find the nth derivative, you can use the formula: f^(n) (x) = (-1)^n * (n!)/(x + 1)^(n+1).