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- Exponential Functions - Introduction
Exponential Functions - Introductions
If $a > 0$ and $a \ne 1$, then the exponential function with base $_a$ is given by $f( x ) = a^x$.
Natural Exponential Functions
1. The irrational number $e = \mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{1}{n}} \right)^n} \approx 2.71828 \cdot \cdot \cdot $ or $e = \mathop {\lim }\limits_{x \to 0} {\left( {1 + x} \right)^{\frac{1}{x}}}$.
2. For each real $x$, $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{x}{n}} \right)^n} = e^x$.
3. $f(x) = e^x$ is a natural exponential function.
Derivatives of Exponential Functions
1. $\frac{d}{{dx}}[{e^x}] = e^x$
2. $\frac{d}{{dx}}[e^{f(x)}] = {e^{f(x)}} \cdot f'(x)$.
The hyperbolic cosine and sine functions
The hyperbolic cosine function is defined as
$\cosh (x) = \frac{{\exp (x) + \exp ( - x)}}{2}$, $- \infty < x < \infty$,
while the hyperbolic sine function is defined as
$\sin (x) = \frac{{\exp (x) - \exp ( - x)}}{2}$, $- \infty < x < \infty$.
Properties of hyperbolic cosine and sine functions
1. $(\sinh (x))' = \cosh (x)$ , $(\cosh (x))' = \sinh (x)$.
2. $\cosh (x) > \sinh (x)$.
3. ${\cosh ^2}(x) - {\sinh ^2}(x) = 1$