General Differentiation Rules

A derivative of a function is the rate of change of the function or the slope of the tangent line at a given point. The derivative of f(a) is notated as

f^{'} (a)

\frac{d}{d x} f (a)

.

This discussion will focus on the Basic Differentiation Rules for differentiating constant, constant multiple, power, exponential and logarithmic functions.

The derivative of a Constant Function is zero. This is because the slope of the tangent line at any point is zero. Think about the function f(x) = 4 or y = 4. This is simply a horizontal line that passes through point (0, 4). At any point on this line the slope of the tangent is zero. In other words, there is no rate of change.

DERIVATIVES OF CONSTANT FUNCTIONS: $\frac{d}{d x} (c) = 0$

FUNCTION	$f (x) = \frac{1}{2}$	f(x) = 359	$f (x) = 4 π^{*}$
DERIVATIVE	$f^{'} (x) = 0$	$f^{'} (x) = 0$	$f^{'} (x) = 0$

* Remember that π is simply an irrational constant.

The derivative of a Function Multiplied by a Constant is equal to the constant multiplied by the derivative of the function. For example if f(x) = 7x the derivative of the function is

7 \frac{d}{d x} x

, read as seven times the derivative of x.

DERIVATIVES OF CONSTANT MULTIPLE FUNCTIONS: $\frac{d}{d x} [c f (x)] = c \frac{d}{d x} f (x)$

FUNCTION	f(x) = 5x²	f(x) = -3 ln x	$f (x) = \frac{4}{3} π^{*} r^{3}$
DERIVATIVE	$f^{'} (x) = 5 \frac{d}{d x} x^{2}$	$f^{'} (x) = - 3 \frac{d}{d x} \ln x$	$f^{'} (x) = \frac{4}{3} π \frac{d}{d x} r^{3}$

* Remember that π is simply an irrational constant.

The derivative of the Power Function, xⁿ, is equal to nx^n-1. Multiply the base of the exponent (x) by the value of the power (n) and subtract one from the power (n - 1).

DERIVATIVES OF POWER FUNCTIONS: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$

FUNCTION

f(x) = x²

f(x) = x^-3

$f (x) = \frac{1}{x^{4}}$

DERIVATIVE

$f^{'} (x) = 2 x^{1}$

$f^{'} (x) = 2 x$

$f^{'} (x) = - 3 x^{- 3 - 1}$

$f^{'} (x) = - 3 x^{- 4}$

$f^{'} (x) = x^{- 4}$

$f^{'} (x) = - 4 x^{- 4 - 1}$

$f^{'} (x) = - 4 x^{- 5}$

The derivative of the Common Exponential Function, a^x, where a is any constant, is equal to (ln a)a^x. Take the natural log of the base of the exponent and multiply it by the exponent.

DERIVATIVES OF COMMON EXPONENTIAL FUNCTIONS: $\frac{d}{d x} (a^{x}) = (l n a) a^{x}$

FUNCTION	f(x) = 100^x	$f (x) = {\sqrt{5}}^{x}$	$f (x) = {(\frac{1}{8})}^{x}$
DERIVATIVE	$f^{'} (x) = (\ln 100) 100^{x}$	$f^{'} (x) = (\ln \sqrt{5}) {\sqrt{5}}^{x}$	$f^{'} (x) = (\ln \frac{1}{8}) {\frac{1}{8}}^{x}$

The natural exponential function is the common logarithmic function when a = e, where e is the irrational number approximated as 2.718. The derivative of the Natural Exponential Function, e^x, is equal to e^x.

DERIVATIVE OF NATURAL EXPONENTIAL FUNCTION: $\frac{d}{d x} (e^{x}) = e^{x}$

The derivative of the Common Logarithmic Function, log_a^x, where a is any constant, is equal to

\frac{1}{{(l n a)}^{x}}

. Take one and divide in by the natural log of the base (a) to the x power.

DERIVATIVES OF COMMON LOGARITHMIC FUNCTIONS: $\frac{d}{d x} l o g_{a} x = \frac{1}{{(l n a)}^{x}}$

FUNCTION	$f (x) = l o g_{5} x$	$f (x) = l o g_{0.2} x$	$f (x) = \log^{*} x$
DERIVATIVE	$f^{'} (x) = \frac{1}{{(l n 5)}^{x}}$	$f^{'} (x) = \frac{1}{{(l n 0.2)}^{x}}$	$f^{'} (x) = \frac{1}{{(l n 10)}^{x}}$

* log with no base is understood to be log₁₀.

The natural logarithmic function is the common logarithmic function when a = e, where e is the irrational number approximated as 2.718. The Natural Logarithmic Function, log_e^x, has the special notation lnx. The derivative of the natural logarithmic function is

\frac{1}{x}

DERIVATIVE OF NATURAL LOGARITHMIC FUNCTION: $\frac{d}{d x} l n x = \frac{1}{x}$

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