Given that y = 4x^3 – 5/(x^2) , x not equal to 0, find in their simplest form (a) dy/dx, and (b) integral of y with respect to x.

a) When differentiating y, the method with each term is to 'times by the power and minus 1'. In order to apply this, we need every term to consist of a coefficient multiplied by a power of x. To start with we therefore need to rewrite the equation as y = 4x^3 - 5x^(-2). So, with the first term '4x^3' we first multiply by 3 and then take away 1 from the power. This gives '34x^2=12x^2'. We repeat with the second term to get '(-2)(-5)x^(-3)'=10x^(-3)'. Finally we rewrite the complete sum in the form the question is given, as dy/dx=12x^2 + 10/(x^3). b) When integrating y, we do the opposite. In other words, we 'add 1 to the power and divide by the new power'. This is easiest to do with the equation in the rewritten form as above. Taking the first term '4x^3' we add 1 to the power to get 4, then divide by this new power. This gives '(4x^4)/4= x^4'. Repeating with the second term we get '(-5x^-1)/(-1)=5x^(-1)'. Finally we rewrite in the original form and add the arbitrary constant 'c', to give 'x^4 + 5/x +c'.