Ho do I use the limit definition of derivative to find f'(x) for f(x)=3x2+x?

Question

Ho do I use the limit definition of derivative to find f&#039;(x) for f(x)=3x2+x?

Aidyn Wall · Accepted Answer

By Power Rule, we know that we are supposed to get f'(x)=6x+1
Let us find it using the definition

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

Let us first find the difference quotient

\frac{f (x + h) - f (x)}{h} = \frac{3 {(x + h)}^{2} + (x + h) - [3 x^{2} + x]}{h}

By simplifying the numerator,

= \frac{3 (x^{2} + 2 x h + h^{2}) + x + h - 3 x - x}{h}

= \frac{3 x^{2} + 6 x h + 3 h^{2} + h - 3 x^{2}}{h}

= \frac{6 x h + 3 h^{2} + h}{h}

By factoring out h from the numerator,

= \frac{h (6 x + 3 h + 1)}{h}

By cancelling out h's,
=6x+3h+1
Hence,

f^{'} (x) = lim_{h \to 0} (6 x + 3 h + 1) = 6 x + 3 (0) + 1 = 6 x + 1

lernarfnincln6g · Accepted Answer

The derivative of f(x) with respect to x is defined as:limh→0[f(x+h)−f(x)h]=limh→0(3(x+h)2+x+h−(3x2+x)h)=limh→0(3(x+h)2+x+h−(3x2+x)h)=6x+1Answer:=6x+1

Ho do I use the limit definition of

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