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Differentiation Rules

The Essentials

Using the limit definition of a derivative is not always time effective and often more complicated than we want it to be. In order to bypass this lengthy process we have developed rules for finding derivatives in a quicker, simpler way.

Let c be any constant, let n be a positive integer, and let $f (x)$ and $g (x)$ be differentiable functions. If these assumptions are true then each of the following rules applies

The Constant Rule

If f (x) = c, then f^{'} (c) = 0

May also be expressed

\frac{d}{d x} (c) = 0

The Power Rule

If $f (x) = x^{n}$ , then

f^{'} (x) = n x^{n - 1}

May also be expressed

\frac{d}{d x} (x^{n}) = n x^{n - 1}

The Sum Rule

\frac{d}{d x} (f (x) + g (x)) = \frac{d}{d x} (f (x)) + \frac{d}{d x} (g (x))

The Difference Rule

\frac{d}{d x} (f (x) - g (x)) = \frac{d}{d x} (f (x)) - \frac{d}{d x} (g (x))

The Constant Multiple Rule

\frac{d}{d x} (k f (x)) = k \frac{d}{d x} (f (x))

The Product Rule

\frac{d}{d x} (f (x) * g (x)) = \frac{d}{d x} (f (x)) * g (x) + \frac{d}{d x} (g (x)) * f (x)

The Quotient Rule

\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{\frac{d}{d x} (f (x)) g (x) - \frac{d}{d x} (g (x)) f (x)}{{(g (x))}^{2}}

A Deeper Dive

Each of these rules applies to certain circumstances found among functions. It should be noted that each of these can be applied in combination one with another. Most complex engineering problems involving derivatives will require the use of more than one differentiation rule. Thus it is important to understand the interaction between the differentiation rules. In most cases we will want to apply the rules in reverse order of how we would evaluate the function.

Practice

Solve the following problems:

(1) Use the product and power rule to find the derivative.

\[ h(x) = (x^2 + 2)(x + 1) \]

2) Use the quotient rule to find the derivative.

\[ m(x) = \frac{2x}{\sin(x)} \]

Solutions:

(1) For this problem lets make $ f(x) = (x^2 + 2) $ and $ g(x) = (x + 1) $. Now it’s time to plug $ f(x) $ and $ g(x) $ into the formula.

$ h'(x) = (2x)(x + 1) + (x^2 + x)(1) $

After simplifying:

\[ h'(x) = 3x^2 + 2x + 2 \]

(2) For this problem, let make $ f(x) = 2x $ and $ g(x) = \sin(x) $. Now we can plug $ f(x) $ and $ g(x) $ into the formula above for the quotient rule.

\[ m'(x) = \frac{(2)\sin(x) - \cos(x)(2x)}{(\sin(x))^2} \]