How do you find the derivative using limits of f(x)=9−12x?

Question

Dallelopeelvep2yc · Accepted Answer

We will use the limit definition, f′(x)=limx→0f(x+h)−f(x)h.f′(x)=limx→09−12(x+h)−(9−12x)hf′(x)=limx→09−12x−12h−9+12xhf′(x)=limx→0−12hhf′(x)=limx→0−12f′(x)=−12

seskew192atp · Accepted Answer

This is the formula for finding the derivative using limits:

\frac{d}{d x} f (x) = lim_{h \to 0} \frac{f (x - h) - f (x)}{h}

Plug in (x-h) wherever you see an x in your f(x) for the first part of your numerator and then just the function itself for the second part of the numerator.
Then, its just basic arithmetic after that:

= lim_{h \to 0} \frac{(9 - \frac{x - h}{2}) - (9 - \frac{x}{2})}{h}

= lim_{h \to 0} \frac{(\frac{18}{2} - \frac{x - h}{2}) - (9 - \frac{x}{2})}{h}

= lim_{h \to 0} \frac{(\frac{18 - x + h}{2}) - (\frac{18 - x}{2})}{h}

= lim_{h \to 0} \frac{\frac{18 - x + h}{2} - \frac{18 + x}{2}}{h}

= lim_{h \to 0} - \frac{1}{2}

= - \frac{1}{2}

How do you find the derivative using limits

Answered question

Answer & Explanation

New Questions in Calculus 1