Given that the curve y = 3x^2 + 6x^1/3 + (2x^3)/3x^1, find an expression for the gradient of the curve.

To find the gradient of a curve, you simply differentiate the equation of the curve. The first thing I like to do in any differentiation question is to simplify each expression where you can i.e. whenever there is a common factor in the numerator and denominator of an expression. In this case, we have (2x^3)/3x^1, which can be re-written as (2/3)(x^3/x^1) this has an x value in the denominator and in the numerator and so to factorise the expression, we can apply the law of division of indices that says that 'When expressions with the same base are divided, the indices are subtracted' hence the simplified expression would be (2/3)(x^2)

In order to differentiate each expressions, you apply the rule of differentiation for y= ax^n, dy/dx(which is the diferential)= anx^n-1. Applying this rule to the question above, the answer derived will be 6x^1 + 2x^-2/3 + (4/3)x^1. Hence, simplyfying the expression, the gradient of the curve, dy/dx = 6x + 2x^-2/3 +4x/3.