How do you find the derivative of $f(x)=ax^2$ ?

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Answer 1 · 2017-01-09T15:45:27

$d/(dx) (ax^2) = 2ax$

This is an application of a basic differentiation rule:

$d/(dx) x^n = n*x^(n-1)$

In this case:

$d/(dx) (ax^2) = a* d/(dx) (x^2) = 2ax$

You can also demonstrate it directly considering the formal definition of the derivative of $f(x)$ :

$f'(x) = lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax) = lim_(Deltax->0) (Deltaf)/(Deltax)$

$Delta f = a(x+Deltax)^2 - ax^2 = ax^2+2axDeltax +a(Deltax)^2-ax^2=2axDeltax +a(Deltax)^2$

$(Deltaf)/(Deltax) = 2ax +aDeltax$

$lim_(Deltax->0) (Deltaf)/(Deltax) = lim_(Deltax->0)2ax +aDeltax = 2ax$

How do you find the derivative of f(x)=ax^2f(x)=ax^2?