If $f ( x ) = - 3 x ^ { 2 } - 4 x$, find the following limit, if it exists.
$$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$$
A) $- 6 x - 4$
B) $- 3 x - 4$
C) $2 x - 4$
D) $- 6 x$
E) The limit does not exist.

Question

Quizplus · Accepted Answer

This limit represents the derivative of the function $f(x) = -3x^2 - 4x$ at any point $x$. Calculating the derivative using the definition of the derivative gives $\frac{d}{dx}(-3x^2 - 4x) = -6x - 4$, which matches choice A.

If f(x)=−3x2−4xf ( x ) = - 3 x ^ { 2 } - 4 xf(x)=−3x2−4x

If $f ( x ) = - 3 x ^ { 2 } - 4 x$