$e^{lnx} = x$

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)$: $f'(x) = {g'(x)}/{g(x)} = {6x}/{3x^2} = 2/x$

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:

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

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$

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 $f'(x) = -1/{1+x^2}$

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) $

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b. 1939

Gro Harlem Brundtland, born 1939, is one of the most important figureheads and leaders of the environmental movement. In 1987, she chaired the famous Brundtland Commission for the UNO, which established Sustainable Development as a central goal of development and environmental protection.

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We have become great because of the lavish use of our resources. But the time has come to inquire seriously what will happen when our forests are gone, when the coal, the iron, the oil and the gas are exhausted, when the soils have still further impoverished and washed into the streams, polluting the rivers, denuding the fields and obstructing navigation.

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