# Derivatives of trigs, exponents and logs

## Logs and Exponents

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\$

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

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) \$

## Site Index

### Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:

Mechanics

and many more articles in the subject:

### Universe

'Universe' on ScienceLibrary.info covers astronomy, cosmology, and space exploration. Learn Science with ScienceLibrary.info.

### Great Scientists

#### Albert Einstein

1879 - 1955

Albert Einstein is considered by many to be the greatest scientist of the 20th century, and his contributions to science equal in importance and scope to those of Isaac Newton.

### Quote of the day...

Nonno's explanations always did a round-robin circuit of science - history - philosophy, then back to science. Always back to science. As if it were that that drove the mechanism of time, and not the other way round. And philosophy? Well, that just went along for the ride.