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A Derivation

Lingual Usage

  1. A leading or drawing off of water from a stream or source.
  2. The act of receiving anything from a source; the act of procuring an effect from a cause, means, or condition, as profits from capital, conclusions or opinions from evidence.
  3. (grammar, genealogy) The act of tracingorigin or descent.
  4. (grammar) Forming a new word by changing the base of another word or by adding affixes to it.
  5. The state or method of being derived; the relation of origin when established or asserted.
  6. That from which a thing is derived.
  7. That which is derived; a derivative; a deduction.
  8. (mathematics) The operation of deducing one function from another according to a fixed definition, referred to as derivation or differentiation; this is the inverse operation to integration.
  9. (medicine) A drawing of humors or fluidsfrom one part of the body to another, to relieve or lessen a morbid process.

Mathematics

Derivation refers to the process of systematically finding the derivative of a function with respect to one or more variables. In calculus, the derivative represents the rate at which a quantity changes with respect to another variable. The result of the derivation is the derivative function, which gives information about the slope or rate of change of the original function.

\[f’(x)=\lim_{\Delta x \to 0} \frac {\Delta f(x)}{\Delta x}\]

\[\Delta x=x-x_0\]

so:

\[\Delta f(x)=f(x)-f(x_0)\]

\[f’(x)=\lim_{\Delta x \to 0} \frac {\Delta f(x)}{\Delta x}=\lim_{ x \to x_0} \frac {f(x)-f(x_0)}{x- x_0}\]

\[\Delta x=x-x_0=\epsilon\]

\[f’(x)=\lim_{ \epsilon\to 0} \frac {f(x+\epsilon)-f(x)}{\epsilon}\]

The process of derivation involves applying various rules and techniques to functions. Some of the key rules include:

  1. Power Rule: If \(f(x) =x^a\), where (a) is a constant, then \(f'(x) = nx^{a-1}\).

\[f’(x)=\lim_{ \epsilon\to 0} \frac {f(x+\epsilon)-f(x)}{\epsilon}\]

\[f’(x)=\lim_{ \epsilon\to 0} \frac {(x+\epsilon)^a-(x)^a}{\epsilon}\]

The expression \((x+\epsilon)^a\) can be expanded using the binomial theorem:

\[(x+\epsilon)^a = x^{a} + \sum_{i=1}^{a} \binom{a}{i} x^{a-i} \epsilon^1\]

\[(x+\epsilon)^a - x^{a} = a x^{a-1} \epsilon^1+\sum_{i=2}^{a} \binom{a}{i} x^{a-i} \epsilon^i\]

\[\frac {(x+\epsilon)^a-(x)^a}{\epsilon}=\frac {a x^{a-1} \epsilon^1+\sum_{i=2}^{a} \binom{a}{i} x^{a-i} \epsilon}{\epsilon}\]

\[\frac {(x+\epsilon)^a-(x)^a}{\epsilon}={a x^{a-1} +\sum_{i=2}^{a} \binom{a}{i} x^{a-i} \epsilon^{i-1}}\]

\[f’(x)={a x^{a-1} +\lim_{ \epsilon\to 0} \{\sum_{i=2}^{a} \binom{a}{i} x^{a-i} \epsilon^{i-1}}\}\]

\(\lim_{ \epsilon\to 0} {\sum_{i=2}^{a} \binom{a}{i} x^{a-i} \epsilon^{i-1}}=0\)because \(i-1>0\)

\[f’(x)=a x^{a-1} \]

  1. Product Rule: If \(f(x) = u(x) \cdot v(x)\), then \(f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)\).

Apply the definition of the derivative on f(x):

\[\frac{df}{dx} = \lim_{h \to 0} \frac{u(x + h)v(x + h) - u(x)v(x)}{h} \]

Using algebra to isolate the terms involving (u) and (v):

\[\frac{df}{dx} = \lim_{h \to 0} \frac{u(x + h)v(x + h) - u(x)v(x) + u(x)v(x + h) - u(x)v(x + h)}{h}\]

Group the terms differently:

\[\frac{df}{dx} = \lim_{h \to 0} \frac{u(x + h)v(x + h) - u(x)v(x + h)}{h} + \frac{u(x)v(x + h) - u(x)v(x)}{h}\]

Now, factor out \(v(x + h)\) from the first term and u(x) from the second term:

\[ \frac{df}{dx} = \lim_{h \to 0}\{ \frac{v(x + h)[u(x + h) - u(x)]}{h} + \frac{u(x)[v(x + h) - v(x)]}{h}\}\]

Now, notice that both terms inside the brackets involve differences that approach zero as (h) approaches zero. This allows us to rewrite the limit as two separate limits:

\[\frac{df}{dx} = \lim_{h \to 0} \frac{v(x + h)[u(x + h) - u(x)]}{h} + \lim_{h \to 0} \frac{u(x)[v(x + h) - v(x)]}{h} \]

\[\frac{df}{dx} = \lim _{h \to 0}v(x + h)\lim_{h \to 0} \frac{u(x + h) - u(x)}{h} +u(x) \lim_{h \to 0} \frac{v(x + h) - v(x)}{h} \]

\[\frac{df}{dx} = v(x)u’(x)+u(x)v’(x)\]

  1. Quotient Rule: If \(f(x) = \frac{g(x)}{h(x)}\), then \(f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2}\).
  1. Chain Rule: If \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x)) \cdot h'(x)\).
  1. Sum/Difference Rule: If \(f(x) = g(x) + h(x)\), then \(f'(x) = g'(x) + h'(x)\).

Derivation is a fundamental concept in calculus and is used to analyze functions, understand their behavior, and solve problems related to rates of change and slopes. The derivative provides a powerful tool for studying the properties of functions and plays a crucial role in many areas of mathematics and science.