rationalskepticism.org

For the following functions, please find the first and second derivatives:

1. [math]f(x)=x^3+2
2. [math]f(x)=5x^7+x^{-5}
3. [math]f(x)=8x^{-3}-9x^7-4
4. [math]f(x)=bx^f+c
   where b,f and c are constants

Let's start solving:

1. We are given the equation [math]f(x)=x^3+2. Remember that the derivative of this is the addition of each term, as each term can be seen as a separate function. So, we start with the first term: [math]x^3. What is its derivative? Remembering the rule, a=1, n=3 and n-1=2. Then the derivative of this is: [math]3x^2. The second term is 2, which is a constant and so its derivative is equal to 0. Adding those two we get the following:
   
   
   [math]f'(x)=3x^2
   
   
   There is no need to add the 0, the derivative of the 2 term.
   
   We also need the second derivative; just follow the same rule and we have: a=3, n=2 and n-1=1. So, the second derivative is:
   
   
   [math]f''(x)=3\times 2\times x
   
   
   [math]f''(x)=6x