# 2.13: Definition of Derivative Examples

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

In the last section, we saw the instantaneous rate of change, or derivative, of a function $$f(x)$$ at a point $$x$$ is given by

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

## Definition of Derivative 1

Find the derivative of the function $$f(x) = 3x + 5$$ using the definition of the derivative.

To use this in the formula $$f'(x) = \frac{f(x+h) - f(x)}{h}$$, first we need to replace the $$f(x+h)$$ part of the formula. This is the same as $$f(x)$$ which is $$3x+5$$, except we replace $$x$$ with that $$(x+h)$$ in parantheses. Like the following. The colors are only to highlight the substitution of $$f(x+h)$$ and $$f(x)$$. We’ll drop the colors as soon as we need to combine expressions.

\begin{align*} f'(x) & = \lim_{h \to 0} \frac ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02:_Derivative_Introduction/2.13:_Definition_of_Derivative_Examples), /content/body/div/div/p/span, line 1, column 1  {h} \\ & = \lim_{h \to 0} \frac ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02:_Derivative_Introduction/2.13:_Definition_of_Derivative_Examples), /content/body/div/div/p/span, line 1, column 1  {h} \\ \end{align*}

Now we continue to simplify and find the answer.

\begin{align*} f'(x) & = \lim_{h \to 0} \frac ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02:_Derivative_Introduction/2.13:_Definition_of_Derivative_Examples), /content/body/div/div/p/span, line 1, column 1  {h} \\ & = \lim_{h \to 0} \frac ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02:_Derivative_Introduction/2.13:_Definition_of_Derivative_Examples), /content/body/div/div/p/span, line 1, column 1  {h} \\ & = \lim_{h \to 0} \frac{3x + 3h + 5 - 3x - 5}{h} \\ & = \lim_{h \to 0} \frac{3h}{h} \\ & = \lim_{h \to 0} 3 \\ & = \boxed{3} \end{align*}

Here, we have $$f'(x) = 3$$. That makes sense if you think about it: $$3x + 5$$ is a line with slope $$3$$!

## Definition of Derivative 2

Find the derivative of $$f(x) = x^2$$ using the definition.

\begin{align*} f'(x) & = \lim_{h \to 0} \frac ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02:_Derivative_Introduction/2.13:_Definition_of_Derivative_Examples), /content/body/div/div/p/span, line 1, column 1  {h} \\ & = \lim_{h \to 0} \frac ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02:_Derivative_Introduction/2.13:_Definition_of_Derivative_Examples), /content/body/div/div/p/span, line 1, column 1  {h} \\ & = \lim_{h \to 0} \frac ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02:_Derivative_Introduction/2.13:_Definition_of_Derivative_Examples), /content/body/div/div/p/span, line 1, column 1  {h} \\ & = \lim_{h \to 0} \frac{2xh + h^2}{h} \\ & = \lim_{h \to 0} \frac{h(2x + h)}{h} \\ & = \lim_{h \to 0} 2x + h \\ & = 2x + (0) \\ & = \boxed{2x} \end{align*}

So what does this mean? Well, this means we double $$x$$ to find the slope of the tangent line of $$f(x) =x^2$$. So at $$x = 3$$, the slope is $$6$$, and at $$x = 1.2$$, the slope is $$2.4$$. ETC.

This page titled 2.13: Definition of Derivative Examples is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.