Use the definition of a derivative, using x and 'change in' x to find the derivative of

Possible derivation:
d/dx(-3 x+4 x^2)
Differentiate the sum term by term and factor out constants:
 = -3 (d/dx(x))+4 (d/dx(x^2))
The derivative of x is 1:
 = 4 (d/dx(x^2))-1 3
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
 = -3+4 2 x
Simplify the expression:
Answer: |= -3 + 8x

The definition of derivative is \(\displaystyle\lim_{\Delta x \rightarrow 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\)

Let f(x) = 4x^2 - 3x.

\(\quad f'(x)\\ =\displaystyle\lim_{\Delta x \rightarrow 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\\ =\displaystyle\lim_{\Delta x \rightarrow 0}\dfrac{(4(x+\Delta x)^2 - 3(x + \Delta x))-(4x^2-3x)}{\Delta x}\\ =\displaystyle\lim_{\Delta x \rightarrow 0}\dfrac{4x^2+4\Delta x^2 + 8x\Delta x - 3x - 3\Delta x-4x^2+3x}{\Delta x}\\ =\displaystyle\lim_{\Delta x \rightarrow 0}\dfrac{4\Delta x^2 + 8x\Delta x - 3\Delta x}{\Delta x}\\ =\displaystyle\lim_{\Delta x \rightarrow 0}4\Delta x + 8x - 3\\ = 4(0) + 8x + 3\\ = 8x + 3\)

Max: The final result should be:8x - 3 not 8x + 3.