Differential Calculus

The slope of a line is defined to be the rise divided by the run. For a curved line, however, the slope has to be defined at a point. Lines (curved or straight, but not infinitely steep) can always be thought of as functions of a single variable. We call the slope of a line evaluated at any given point its derivative, and call the process of finding that slope taking the derivative of the function.

Later we'll say a few words about multivariate (vector) differential calculus, but that is mostly beyond the scope of this course.

The definition of the derivative of a function is:

$$\frac{d f}{d x} = \lim_{\Delta x \to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$$ (86)

This is the slope of the function at the point $x$ .

First, note that:

$$\frac{d (a f)}{d x} = a \frac{d f}{d x}$$ (87)

for any constant $a$ . The constant simply factors out of the definition above.

Second, differentiation is linear. That is:

$$\frac{d }{d x}(f(x)+g(x)) = \frac{d f(x)}{d x} + \frac{d g(x)}{d x}$$ (88)

Third, suppose that $f = g h$ (the product of two functions). Then

$$\frac{d f}{d x} = \frac{d (gh)}{d x} = \lim_{\Delta x \to 0}\frac{g(x+\Delta x)h(x + \Delta x) - g(x)h(x)}{\Delta x}$$

$$= \lim_{\Delta x \to 0}\frac{\left(g(x) + \frac{d g}{d x}\Delta x)(h(x) + \frac{d h}{d x}\Delta x) - g(x)h(x) \right)}{\Delta x}$$

$$= \lim_{\Delta x \to 0}\frac{\left(g(x)\frac{d h}{d x}\Delta x + \... ...x}h(x)\Delta x + \frac{d g}{d x}\frac{d h}{d x}(\Delta x)^2)\right)} {\Delta x}$$

$$= g(x)\frac{d h}{d x} + \frac{d g}{d x}h(x)$$ (89)

where we used the definition above twice and multiplied everything out. If we multiply this rule by $dx$ we obtain the following rule for the differential of a product:

$$d(gh) = g\ dh + h\ dg$$ (90)

This is a very important result and leads us shortly to integration by parts and later in physics to things like Green's theorem in vector calculus.

We can easily and directly compute the derivative of a mononomial:

$$\frac{d x^n}{d x} = \lim_{\Delta x \to 0}\frac{x^n + nx^{n-1}\Delta x + n(n-1)x^{n-2}(\Delta x)^2 \ldots + (\Delta x)^n) - x^2}{\Delta x}$$

$$= \lim_{\Delta x \to 0}\left(nx^{n-1} + n(n-1)x^{n-2}(\Delta x) \ldots + (\Delta x)^{n-1}\right)$$

$$= n x^{n-1}$$ (91)

or we can derive this result by noting that $\frac{d x}{d x} = 1$ , the product rule above, and using induction. If one assumes $\frac{d x^n}{d x} = nx^{n-1}$ , then

$$\frac{d x^{n+1}}{d x} = \frac{d (x^{n}\cdot x)}{d x}$$

$$= nx^{n-1}\cdot x + x^n\cdot 1$$

$$= nx^n + x^n = (n+1)x^n$$ (92)

and we're done.

Again it is beyond the scope of this short review to completely rederive all of the results of a calculus class, but from what has been presented already one can see how one can systematically proceed. We conclude, therefore, with a simple table of useful derivatives and results in summary (including those above):

$$\frac{d a}{d x} = 0 \quad\quad a\ {\rm constant}$$ (93)

$$\frac{d (a f(x)}{d x} = a\frac{d f(x)}{d x} \quad\quad a\ {\rm constant}$$ (94)

$$\frac{d x^n}{d x} = n x^{n-1}$$ (95)

$$\frac{d }{d x}(f(x)+g(x)) = \frac{d f(x)}{d x} + \frac{d g(x)}{d x}$$ (96)

$$\frac{d f}{d x} = \frac{d f}{d u} \frac{d u}{d x} \quad\quad {\rm chain\ rule}$$ (97)

$$\frac{d (gh)}{d x} = g\frac{d h}{d x} + \frac{d g}{d x}h \quad\quad {\rm product\ rule}$$ (98)

$$\frac{d (g/h)}{d x} = \frac{\frac{d g}{d x}h - g \frac{d h}{d x}}{h^2}$$ (99)

$$\frac{d e^x}{d x} = e^x$$ (100)

$$\frac{d e^{(ax)}}{d x} = ae^x \quad\quad {\rm from\ chain\ rule,\ } u = ax$$ (101)

$$\frac{d \sin(ax)}{d x} = a\cos(x)$$ (102)

$$\frac{d \cos(ax)}{d x} = -a\sin(x)$$ (103)

$$\frac{d \tan(ax)}{d x} = \frac{a}{\cos^2(ax)} = a \sec^2(ax)$$ (104)

$$\frac{d \cot(ax)}{d x} = -\frac{a}{\sin^2(ax)} = -a\csc^2(ax)$$ (105)

$$\frac{d \ln(x)}{d x} = \frac{1}{x}$$ (106)

There are a few more integration rules that can be useful in this course, but nearly all of them can be derived in place using these rules, especially the chain rule and product rule.