How do I find the 2nd derivative of x^(-1/15)

d/dx x^(-1/15) = (-1/15)x^(-16/15)

(-1/15)d/dx x^(-16/15) = (16/225)x^-(31/15)

To find the second derivative of a function, we need to start by finding the first derivative of the function, and then find the derivative of that first derivative.

Let's start by finding the first derivative of the function f(x) = x^(-1/15).

Step 1: Take the natural logarithm of both sides of the equation to simplify the expression:
ln(f(x)) = ln(x^(-1/15))

Step 2: Apply the logarithm rule for exponents to simplify further:
ln(f(x)) = (-1/15) * ln(x)

Step 3: Differentiate both sides of the equation using the chain rule. Let's denote the derivative of f(x) as f'(x):
f'(x)/f(x) = (-1/15) * (1/x)

Step 4: Multiply both sides by f(x) to isolate f'(x):
f'(x) = (-1/15) * (1/x) * f(x)
= (-1/15x) * f(x)
= (-1/15x) * x^(-1/15)

Now, let's find the second derivative by differentiating the expression we obtained for f'(x).

Step 5: Apply the product rule to differentiate f'(x) = (-1/15x) * x^(-1/15):
f"(x) = (-1/15)(-1/15x)(-1/15) * x^(-1/15 - 1)
= (1/225x^2) * x^(-16/15)

Step 6: Simplify the expression:
f"(x) = (1/225x^2) * x^(-16/15)
= 1/225 * x^[-(2+16)/15]
= 1/225 * x^(-18/15)
= 1/225 * x^(-6/5)

Therefore, the second derivative of f(x) = x^(-1/15) is f"(x) = 1/225 * x^(-6/5).

To find the second derivative of the function f(x) = x^(-1/15), you'll first need to find the first derivative and then differentiate it again.

Step 1: Find the first derivative (f'(x)):
To find the first derivative, apply the power rule for differentiation. According to the power rule, if you have a function of the form f(x) = x^n, then its derivative is f'(x) = n * x^(n-1). Applying this rule to our function f(x) = x^(-1/15), we have:

f'(x) = (-1/15) * x^((-1/15) - 1) = (-1/15) * x^(-16/15)

Step 2: Find the second derivative (f''(x)):
To find the second derivative, you need to differentiate the first derivative obtained in Step 1. Here, we apply the power rule again:

f''(x) = d/dx [(-1/15) * x^(-16/15)]
= (-1/15) * (-16/15) * x^(-16/15 - 1)
= (16/225) * x^(-31/15)

Therefore, the second derivative of f(x) = x^(-1/15) is f''(x) = (16/225) * x^(-31/15).