Find the derivative of x³ with these two methods

In GCSE Maths, when being asked the derivative of x³, it is important to not overcomplicate it.

The derivative of x³ can be a very simple question if you have practice using the power rule which can be applied to polynomials of the form axˆn.

The derivative of axⁿ using the formula is anx^n-1

In this case where n=3 and a=1. 

The derivative of x³ is 3x²

Solving the derivative from first principles

Solving the derivative of x³ from first principles:

The first principles formula is dy/dx = [f(x+h)-f(x)]/h  as h approaches 0.

Let f(x)=x³, and f(x+h)=(x+h)³.

Next we need to expand the (x+h) bracket and we can use binomial expansion here.

(x+h)³ = x⁰h³ + 3x¹h² + 3x²h¹ + x³h⁰

Which simplifies to h³ + 3xh² + 3x²h + x³

Now we make the substitution into the formula.

[h³ + 3xh² + 3x²h + x³ - x³]/h

We are left with [h² + 3xh +3x²]

As h tends to 0, we are left with 3x². This proves that 3x² is the derivative of x³

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