Example #1

Find the derivatives of the following function from first principles: $f\left(x\right)=x^2$.

First calculate f(x+h):

•        we know that $f\left(x\right)=x^2 \Rightarrow f\left(x+h\right)={\left(x+h\right)}^2$
•        expand ${\left(x+h\right)}^2 \Rightarrow x^2+2xh+h^2$

Substitute f(x+h) and f(x) into the formula:

$f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{x^2+2xh+h^2-x^2}{h}$

Simplify: $x^2$ and $-x^2$ cancel each other, therefore then we can divide both the top and bottom by h:

$f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{x^2+2xh+h^2-x^2}{h}=\underset{h\to 0}{\lim }\frac{2xh+h^2}{h}=\underset{h\to 0}{\lim }2x+h$

Then since h heads towards 0, we are free to let $h\to 0$:

$f\text{'}\left(x\right)=\underset{h\to 0}{\lim }2x+h=2x$

So the derivative of f(x) is  f’(x)=2x.

Example #2

Find the derivatives of the following function from first principles: $f\left(x\right)=x^2+6x+9$.

First calculate f(x+h):

•   we know that $f\left(x\right)=x^2+6x+9$$\Rightarrow f\left(x+h\right)={\left(x+h\right)}^2+6\left(x+h\right)+9$

•   expand ${\left(x+h\right)}^2+6\left(x+h\right)+9$$\Rightarrow x^2+2xh+h^2+6x+6h+9$

Substitute f(x+h) and f(x) into the formula:

$f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{x^2+2xh+h^2+6x+6h+9-x^2-6x-9}{h}$

Simplify: $x^2$ and $-x^2$, $6x$ and $-6x$, $9$ and $-9$ cancel each other, then we can divide both the top and bottom by h

$\underset{h\to 0}{\lim }\frac{2xh+h^2+6h}{h}=\underset{h\to 0}{\lim }2x+h+6$

Then since h heads towards 0, we are free to let $h\to 0$:

$\underset{h\to 0}{\lim }2x+h+6=2x+6$

So the derivative of f(x) is  f’(x)=2x+6.

Example #1

Find the derivative of the following function: $f\left(x\right)=2x^3+\frac{4}{5}{x}^2+6x+3$

Differentiate each term separately then and add up the results.

$f\text{'}\left(x\right)=2\times 3x^2+\frac{4}{5}\times 2x+6+0=6x^2+\frac{8}{5}x+6$

Example #2

Find the derivative of the following function: $g\left(x\right)=\frac{1}{\sqrt[3]{x}}$.

First use the laws of exponents to rewrite the function in the form $x^n$: 

$\frac{1}{\sqrt[3]{x}}=\frac{1}{x^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}=x^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Then use the differentiation formula:

$g\text{'}\left(x\right)=\frac{-1}{3}{x}^{\left(\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)-1}=\frac{-1}{3}{x}^{\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Example #3

Find the derivative of the following function: $h\left(x\right)=4e^x+\tan \left(x\right)$.

Differentiate each term separately then and add up the results.

$h\text{'}\left(x\right)=4e^x+\frac{1}{{\cos }^{2}x}$

Example #4

Find the derivative of the following function: $i\left(x\right)=5\sin \left(x\right)+4\cos \left(x\right)$.

Differentiate each term separately then and add up the results.

$i\text{'}\left(x\right)=5\cos \left(x\right)-4\sin \left(x\right)$