Find f''(x), Given that f(x)=5x^3 - 6x^(4/3) + 2x - 3

In order to get from f(x) to f''(x) we need to differentiate the function f(x) with respect to x and then differentiate the resulting function with respect to x again. When differentiating f(x), split the equation f(x) into smaller parts. These parts should be separated by + and - signs. In this case 5x^3 will be one part, -6x^(4/3) will be another then +2x and finally -3. Next step would be to differentiate each part separately, then in the end add them together. Following the rules of differentiation, 5x^3 will become 15x^2, as the constant - number that is not x (in this case 5) - is multiplied by the power of x (which is 3) and then power is reduced by 1 to give us 15x^2. Next part, which is -6x^(4/3) will differentiate to -8x^(1/3). Following the rules of differentiation: -64/3 = -8; 4/3 - 1 = 1/3 Third part, which is +2x, will differentiate into +2. That's because +2x can be written as +2x^1. So by following the rules we get +2x^0. Since any number to the power of 0 equals to 1, +2x^0 can be written as +21 which is +2. Next part would have been -3, but because it is a constant (it does not contain x) when differentiating it will become 0. to finish the first step, we need to add all differentiated parts together, giving 15x^2 - (9/2)x^(-1/4) + 2. This is now f'(x). To get to the f''(x), the f'(x) function must be differentiated again with respect to x. Just like before break the equation into parts, separated by + or - sign and differente each of them separately. 15x^2 will become 30x -(8)x^(1/3) will become -(8/3)x^(-2/3). Be careful with the signs when differentiating, as -8 multiplied by 1/3 will give -8/3 and (1/3)-1 will give -2/3. Next part which is +2 disappears when differentiating as now it is just a constant, so just like -3 in original equation it will become 0. Finally, add all differentiated parts together to get 30x - 8/3 x^(-2/3). With questions like that it is very easy to get signs wrong when differentiating. The best way to make sure this does not happen is to practice these types of questions and not rush it.