Calculus I - The Derivative

Definition

For a real-valued function $f$ of a single real variable, the derivative of $f$ is the function $f'$ whose value at $x$ is

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

if this limit exists. Geometrically, the derivative is the slope of the secant line beween the points $(x, f(x))$ and $(x + h, f(x + h))$. As $h \to 0$, this secant line becomes tangent to the graph of $f$ at the point $(x, f(x))$.

Computing the Derivative By Hand

Although students usually end up memorizing the derivatives for standard, common functions, it is instructive to understand that the derivative can be computed directly from the definition. As an example, let's compute the derivative of $x^3$.

\[ f'(x) \]	=	\[ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]
	=	\[ \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} \]
	=	\[ \lim_{h \to 0} \frac{x^3 +3x^2h + 3xh^2 + h^3 - x^3}{h} \]
	=	\[ \lim_{h \to 0} \frac{3xh(x + h) + h^3}{h} \]
	=	\[ \lim_{h \to 0} 3x(x+h) + h^2 \]
	=	\[ 3x^2 \]

Alternative Notations

The Leibniz notation, $\frac{df}{dx}$ is sometimes used instead of the prime notation to denote the derivate of $f$ with respect to $x$. More specifically, $\frac{d}{dx}$ is thought of as the differentiation operator that is applied to the function $f$.

General Rules

The following rules aid in calculating derivatives:

Sum

\[ (f + g)'(x) = f'(x) + g'(x) \]

Scalar Multiplier

\[ (kf)'(x) = kf'(x) \mbox{, where $k$ is a constant} \]

Product

\[ (fg)'(x) = f(x)g'(x) + f'(x)g(x) \]

Quotient

\[ \frac{f}{g}'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2} \]

Differentials

In Leibniz notation, the chain rule is

\[ \frac{d}{dx} [f(u(x))] = \frac{df}{du}\frac{du}{dx} \]

If we ignore the fact that $\frac{d}{dx}$ is really an operator, and treat it like a fraction, we can multiply the equation by $dx$, giving

\[ d(f(u(x)) = \frac{df}{du} du \]

The above notation of $d$ expression is referred to as a differential. It is interpreted as an infinitely small change to the variable or expression to which it is applied.

Standard Derivatives

Here are the derivatives for a few common functions, expressed in terms of differentials.

\[ d(k) = 0 \] \[ d(u^k) = ku^{k-1}du \] \[ d(e^u) = e^u du \] \[ d(a^u) = (\log a) a^u du \] \[ d(\log u) = \frac{1}{u} du \] \[ d(log_a u) = \frac{1}{u\log a} du \] \[ d(\sin u) = \cos u du \] \[ d(\cos u) = - \sin u du \] \[ d(\tan u) = \sec^2u du \] \[ d(\cot u) = -\csc^2 u du \] \[ d(\sec u) = \sec u \ tan u du \] \[ d(\csc u) = -\csc u \cot u du \] \[ d(\arcsin u) = \frac{du}{\sqrt{1 - u^2}} \] \[ d(\arctan u) = \frac{du}{1 + u^2} \]