Science Library - free educational site

Derivatives of trigs, exponents and logs

Logs and Exponents

Natural log
Natural log and its first differential 1/x

$e^{lnx} = x$

ln$e^x = x$

If $f(x) = e^x$, $f'(x) = e^x$

If $f(x) = e^{g(x)}$, $f'(x) = g'(x)e^{g(x)}$

If $f(x) = $ln$x$, $f'(x) = 1/x$

If $f(x) = a^x$, $f'(x) = ($ln$a)a^x$

For $f(x) = a^x$, $f'(x)$ at $x=0$ is ln$a$.

If $f(x) = $log$_{a}x$, $f'(x) = 1/{xlna}$

If $f(x) = $ln$g(x)$, $f'(x) = {g'(x)}/{g(x)}$

Example: find the derivative of $f(x) = e^{3x^2}$:

$f'(x) = g'(x)e^{g(x)} = 6xe^{3x^2}$

Example: find the derivative of $f(x) = $ln$(3x^2)$: $f'(x) = {g'(x)}/{g(x)} = {6x}/{3x^2} = 2/x$

e exponent x is its own derivative

The letter $e$ was selected to represent the irrational number 2.718281828459, in honour of the great Swiss mathematician Leonhard Euler (1706 - 1781).

$e$ is defined as the limit of $(1 + 1/n)^n$ as n approaches ∞.

It may also be expressed as:

$$e = ∑↙{n=0}↖{∞} 1/{n!} = 1 + 1/1 + 1/{1⋅2} + 1/{1⋅2⋅3} + ...$$

Derivatives of Trigonometric Functions

Sine and cosine

Observing the graph of sine of $x$, it can be seen that the slope is graphed by cosine $x$. When $x$ is zero, sine is zero, with increasing tangent gradient. At zero, the slope is maximum, and gradually decreases, but not constantly, to where it levels out at $π/2$, where the tangent gradient is zero.

cosine is therefore the first derivative of the sine function.

Since the sine and cosine graphs superimpose if sine is translated $π/2$ to the left, cos$x = $sin$(x+π/2)$.

$d/{dx}($cos$x) = d/{dx}[$sin$(x+ π/2)] $

$= [$cos$(x+π/2)]⋅1 = $cos$(x+π/2)$

cos$(x+π/2) = -$sin$x$.

The derivative of $f(x) = sinx$ from first principles:
$f'(x) = {lim}↙{h→0} {f(x+h) - f(x)}/h$
     $= {lim}↙{h→0} {sin(x+h) - sin(x)}/h$
     $= {lim}↙{h→0} ({sin x⋅cos h + cos x ⋅ sin h - sin x}/h)$
     $= {lim}↙{h→0} ({cos x⋅ {sinh}/h + sinx⋅ {cosh -1}/h)$
     $= cosx⋅{lim}↙{h→0} {sinh}/h + sinx⋅{lim}↙{h→0} {cosh - 1}/h$
     $= cosx⋅1 + sinx⋅0 = cosx$

The derivative of tan$x$ can also be derived by a similar procedure:

$d/{dx}($tan$x) = d/{dx}({sinx}/{cosx}) = {cosx(cosx) - sinx(-sinx)}/{(cosx)^2} = {cos^2x+sin^2x}/{cos^2x} = 1/{cos^2x}$, cos$x ≠0$

Basic Trig Differentials

If $f(x) = $sin$ x$, then $f'(x) = $cos$ x$

If $f(x) = $cos$ x$, then $f'(x) = -$sin$ x$

If $f(x) = $tan$ x$, then

$f'(x) = $se$c^2 x$ $ ( = 1/{cos^2x}) = 1 + $tan$^2x$

Basic Trig Differentials

If $f(x) = $arcsin$ x$, then $f'(x) = 1/{√{1-x^2}}$

If $f(x) = $arccos$ x$, then $f'(x) = -1/{√{1-x^2}}$

If $f(x) = $arctan$ x$, then $f'(x) = 1/{1+x^2}$

If $f(x) = $arccot$ x$, then $f'(x) = -1/{1+x^2}$

further Trig Differentials

If $f(x) = $sin$ {x/2}$, then $f'(x) = {1/2}$cos$ {x/2}$

If $f(x) = $cos$ 3x$, then $f'(x) = -3$sin$ 3x$

If $f(x) = $sin$(2x - 1)$, then $f'(x) = 2$cos$(2x - 1)$

If $f(x) = $cot$x$, then

$f'(x) = -$csc$^2x = -1/{sin^2(x)} = -1 - $cot$^2(x) $

Content © All rights reserved. Created : January 23, 2015

Latest Item on Science Library:

The most recent article is:


View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

Subject of the Week


Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on

Gravity lens

Great Scientists

Georges Lemaître

1894 - 1966

Georges Lemaître, 1894 - 1966, was a Belgian astrophysicist, best known for being the originator of the Expanding Universe theory, and the Big Bang origin of the universe.

Georges Lemaître
Transalpine traduzioni

Quote of the day...

Life under capitalism is very, very violent. Even if we're not killing each other in the streets. But we create constant suffering for people.

ZumGuy Internet Promotions

IT information forum by Sean Bone