Differentiation of standard functions

Summary and examples

In general the following derivatives of standard functions are assumed to be known.

	$f(x)$	$f'(x)$

a.	$x^n$	$nx^{n-1}$
b.	$c$	$0$
c.	$\sin(x)$	$\cos(x)$
d.	$\cos(x)$	$-\sin(x)$
e.	$e^x$	$e^x$
f.	$\ln(x)$	$\displaystyle\frac{1}{x}$
g.	$a^x$	$a^x\ln(a)$
h.	$\log_g(x)$	$\displaystyle\frac{1}{x\ln(g)}$

Furthermore we have the following rules:

If:

i. $y=cf(x)$

then:

$y'=cf'(x)$

and if:

j. $y=af(x)+bg(x)$

then:

$y'(x)=af'(x)+bg'(x)$

In the following examples we show how these rules are applied.

Example 1

$y=x^3$

so:

$y'=3x^2$

(according to a)

Example 2

$y=5x^2$

so:

$y'=5.2x=10x$

(according to a, i)

Example 3

$y=7$

so:

$y'=0$

(according to c)

Example 4

$y=8e^x$

so:

$y'=8e^x$

(according to e, i)

Example 5

$y=9\ln(x)$

so:

$y'=\displaystyle\frac{9}{x}$

(according to f, i)

Example 6

$y=4x^2+2\log_7(x)$

so:

$y'=8x+\displaystyle\frac{2}{x}\displaystyle\frac{1}{\ln(7)}$

(according to a, h, i, j)

Example 7

$y=8\ln(x)+4x^2$

so:

$y'=\displaystyle\frac{8}{x}+8x$

(according to a, f, i, j)

Now some more difficult examples.

Example 8

Find the derivative of:

$y=\sqrt[4]{x^3}$

At first sight this function is not in the table of standard functions. However we can rewrite the function:

$y=\sqrt[4]{x^3}=x^{\frac{3}{4}}$

and thus rule a. can be applied with:

$n=\displaystyle\frac{3}{4}$

So:

$y'=\displaystyle\frac{3}{4}x^{\frac{3}{4}-1}=\displaystyle\frac{3}{4\sqrt[4]{x}}$

Example 9

Differentiate:

$y=\displaystyle\frac{9}{5x^3}$

This function can be written as:

$y=\displaystyle\frac{9}{5}x^{-3}$

and thus:

$y'=\displaystyle\frac{9}{5}.-3x^{-4}=-\displaystyle\frac{27}{5x^4}$

(according to a,i)

Example 10

A difficult function seems:

$y=\displaystyle\frac{5x^4+x}{x^4}$

but also this function can be rewritten and be differentiated easily:

$y=\displaystyle\frac{5x^4+x}{x^4}=\displaystyle\frac{5x^4}{x^4}+\displaystyle\frac{x}{x^4}=5+\displaystyle\frac{1}{x^3}=5+x^{-3}$

and thus:

$y'=0-3x^{-4}=-\displaystyle\frac{3}{x^4}$

(according to a, b, j)

Example 11

Differentiate the following function:

$y=x^4-\displaystyle\frac{1}{x^4}$

We can rewrite the function as follows:

$y=x^4-x^{-4}$

and thus:

$y'=4x^3+4x^{-5}=4x^3+\displaystyle\frac{4}{x^5}$

(according to a, i)

Example 12

Finally we want to differentiate the following function:

$y=\displaystyle\frac{1}{\sqrt[5]{x^2}}$

We can rewrite it as follows:

$y=\displaystyle\frac{1}{x^{\frac{2}{5}}}=x^{-\frac{2}{5}}$

The derivative is:

$y'=-\displaystyle\frac{2}{5}x^{-\frac{7}{5}}=-\displaystyle\frac{2}{5x\sqrt[5]{x^2}}$