25  The tools of calculus

In this section, we collect some of the basic facts about manipulating functions.

25.1 Differentiation

Differentiation is craft. Integration is art.

We will often use the Leibniz’ notation for a derivative \[f'(x) = \frac{\mathrm{d}}{\mathrm{d}x} f(x) = \frac{\mathrm{d}f(x)}{\mathrm{d}x} = \frac{\mathrm{d}f}{\mathrm{d}x}.\] This is indicative of a fraction of infinitesimals, and in fact sometimes they do behave like fractions …

Product rule: \[(fg)' = f' g + f g'\] Chain rule: \[f(x(y))' = f'(x(y))y'(x)\] Using Leibniz, \[\frac{\mathrm{d}}{\mathrm{d}x} f(y(x)) = \frac{\mathrm{d}f}{\mathrm{d}y}\frac{\mathrm{d}y}{\mathrm{d}x}.\] Note “cancellation.” Helpful mnemonic device.

Logarithmic differentiation: Sometimes the logarithm of a function is easier to differentiate than the original function. We can then use \[f' = f \cdot (\ln f)'.\]

25.2 Integration

An antiderivative of a function \(f : I \to \mathbb{R}\) is a function \(F : I \to \mathbb{R}\) such that \(F' = f\). Since the derivative of a constant is zero, such can be added freely. Another notation for the antiderivative is the indefinite integral, \[F(x) = \int f(x)\, \mathrm{d}x + C,\] with \(C\) being a constant.

Antiderivatives are tabulated. But there are basic tricks to evaluate your own.

Variable substitution: Write \(x = x(y)\), a function of another variable \(y\). \[\int_{x_0}^{x_1} f(x) \, \mathrm{d}x = \int_{y_0}^{y_1} f(x(y)) \, \frac{\mathrm{d}x}{\mathrm{d}y} \, \mathrm{d}y.\] Here, \(x_i = x(y_i)\). Thus, one must be able to invert the function \(y(x)\) to get \(x(y)\).

Note how the denominator is seemingly “canceled.” Helpful mnemonic device.

Integration by parts: \[\int u'(x) v(x) \, \mathrm{d}x = u(x) v(x) - \int u(x) v'(x) \, \mathrm{d}x.\] Again, an arbitrary constant can be added.

Feynman’s differentiation under the integral sign: Let \(g(x,\alpha)\) be a function that depends on some parameter \(\alpha\) in a smooth manner. Suppose that our function to be integrated i \[f(x) = \frac{\partial}{\partial \alpha} g(x,\alpha),\] for some value \(\alpha\). Under mild conditions on \(g\), \[\int f(x) \, \mathrm{d}x = \int \frac{\partial}{\partial \alpha} f(x,\alpha) \, \mathrm{d}x = \frac{\partial}{\partial \alpha} \int f(x,\alpha) \, \mathrm{d}x.\] The latter integral may be easier to compute.

Example: Gaussian integral.