$e^{lnx} = x$

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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Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on ScienceLibrary.info.

1939

Ernst Ulrich von Weizsäcker, b. 1939, is a prominent German scientist and politician. He is a popular author, with best-selling books like 'Factor Four', the update to the Club of Rome 'Limits to Growth', which explains how a sustainable economy requires reductions in consumption and increases in efficiency of this order of magnitude.

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