# Find The Derivative Of x³ With These Two Methods

When being asked the derivative of x^{3}, it is important to not overcomplicate it.

The derivative of x^{3} 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 ax^{n} using the formula is anx^{n-1}

In this case where n=3 and a=1.

The derivative of x^{3} is 3x^{2}

Solving the derivative of x^{3} from first principles:

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

Let f(x)=x^{3}, 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} = x^{0}h^{3} + 3x^{1}h^{2} + 3x^{2}h^{1} + x^{3}h^{0}

Which simplifies to h^{3} + 3xh^{2} + 3x^{2}h + x^{3}

Now we make the substitution into the formula.

[h^{3} + 3xh^{2} + 3x^{2}h + x^{3} - x^{3}]/h

We are left with [h^{2} + 3xh +3x^{2}]

As h tends to 0, we are left with 3x^{2}. This proves that 3x^{2} is the derivative of x^{3}

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