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Differentiation and integration

Some rules for differentiation

Some simple rules are:

\[\frac{d}{dx} a = 0\]
\[\frac{d}{dx} (a x) = a\]
\[\frac{d}{dx} x^n = n x^{n – 1}\]
\[\frac{d}{dx} log_e(x) = \frac{1}{x}\]
\[\frac{d}{dx} e^x = e^x\]

And some rules about dealing with differentiating functions, sums and products. For functions of \(x\), \(F(x)\) and \(L(x)\):

\[\frac{d}{dx} e^{F(x)} = \frac{dF(x)}{dx} e^{F(x)} = F’(x) e^{F(x)}\]
\[\frac{d}{dx} (F(x)+L(x)) = F’(x) + L’(x)\]
\[\frac{d}{dx} (F(x) . L(x)) = L(x).F’(x) + F(x).L’(x)\]
\[\frac{d}{dx} \left(\frac{F(x)}{L(x)}\right) = \frac{(F'(x))}{(L(x))} - \frac{(F(x).L'(x))}{([L(x)]^2)}\]

or

\[\frac{d}{dx} \left(\frac{F(x)}{L(x)}\right) = \frac{(F'(x)L(x) - L'(x)F(x))}{(L(x)^2)}\]

In general, the chain rule is very helpful.

\[\mbox{The chain rule}: \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\]

Note on partial differentiation

Suppose we have a function, \(F\), of two variables, \(x\) and \(y\).

We can differentiate the function \(F\) with respect to \(x\), \(\frac{\partial F}{\partial x}\). This means we differentiate \(F\) with respect to \(x\) and consider all other variables (i.e. \(y\)) to be fixed.

Similarly, we can obtain the partial derivative of \(F\) with respect to \(y\), \(\frac{\partial F}{\partial y}\).

There are now various second derivatives. We can differentiate with respect to \(x\) twice, \(\frac{\partial^2 F}{\partial x^2}\), differentiate with respect to \(y\) twice, \(\frac{\partial^2 F}{\partial y^2}\), or differentiate with respect to \(x\) and then differentiate with respect to \(y\), \(\frac{\partial^2 F}{\partial x \partial y}\). Doing the last steps in the opposite order gives \(\frac{\partial^2 F}{\partial y \partial x}\).

Integration

Some important integrals. Note, the constant is omitted below, for brevity.

\[\int a \, dx = ax\]
\[\int a F(x) dx = a \int F(x) dx\]
\[\int x^n dx = \frac{x^{n+1}}{n+1}\]
\[\int F’(x) [F(x)]^n dx = \frac{[F(x)]^{(n+1)}}{(n+1)}\]
\[\int e^x dx = e^x\]
\[\int F’(x) e^{F(x)} dx = e^{F(x)}\]
\[\int \frac{1}{x} = log_e (x)\]
\[\int \frac{F'(x)}{F(x)} dx = log_e (F(x))\]
\[\int (F(x) + L(x)) dx = \int F(x) dx + \int L(x) dx\]

Integrating by parts:

\[\int_a^b u \frac{dv}{dx}dx= [uv]_a^b - \int_a^b v \frac{du}{dx}dx\]
Functions, exponential and logarithm rules Matrices