Find the derivative of x³ with these two methods
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 anxn-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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