# Calculus crash course

1. May 31, 2010

### Ben09

I'm a college student needing to take calculus 2 this fall. However, I have not taken calc 1, so since I'm good at teaching myself I'm planning on giving myself a crash course in the material covered in calc 1 over this summer. Can anyone recommend a good calculus book for someone who's fairly good at math but has limited experience with calc? Thanks!

2. Jun 1, 2010

### sEsposito

Hm.. It's going to be pretty hard for you to excel in calc II without taking calc I, frankly, I'm not sure how you even got enrolled into a calc II class without having calc I under your belt, but it's not my business.

There's a book called the calculus life saver, which may be of interest to you. Hope this helps.

3. Jun 1, 2010

### Dickfore

PROTIP:
Rules for differentiation:
1. derivative of a constant
In general, if $f(.)$ does not depend explicitly on some variable, say $x$ it's derivative is zero:
$$\frac{d}{d x}\left(C\right) = 0$$
2. derivative with respect to the argument:
$$\frac{d x}{d x} = 1$$
3. rule of sums
$$\frac{d}{d x}\left[ f(x) + g(x) \right] = \frac{d f(x)}{dx} + \frac{d g(x)}{dx}$$
4. product tule
$$\frac{d}{d x}\left[ f(x) \cdot g(x) \right] = \frac{d f(x)}{dx} \cdot g(x) + f(x) \cdot \frac{d g(x)}{dx}$$
5. chain rule
$$\frac{d}{d x} \left( f[g(x)] \right) = \left. \frac{d f(u)}{du} \right|_{u = g(x)} \cdot \frac{d g(x)}{d x}$$
6. derivative of the exponential function
$$\frac{d \exp(x)}{dx} = \exp(x)$$

Using the above, see if you can derive the following:
1. Quotient rule
$$\frac{d}{d x}\left( \frac{f(x)}{g(x)}\right) = \frac{f'(x) \, g(x) - f(x) \, g'(x)}{[g(x)]^{2}}$$
2. Derivative of a power function:
$$\frac{d}{d x}\left( x^{\alpha} \right) = \alpha \, x^{\alpha - 1}, \ \alpha \in \mathbf{R}$$
3. Derivative of an inverse function
$$y = f(x) \Rightarrow x = f^{-1}(y)$$

$$f[f^{-1}(x)] = x$$

$$\frac{d}{d x}\left( f^{-1}(x) \right) = \frac{1}{f'[f^{-1}(x)]}$$
4. Derivative of a logarithm
$$(\log_{a} {x})' = \frac{1}{x \, \ln{a}}$$
5. Derivative of trigonometric functions
Using Euler's identity:
$$e^{\textup{i} \, x} = \cos{x} + \textup{i} \, \sin{x}$$

and taking the real and imaginary part of the derivative, prove:
$$\begin{array}{l} (\cos{x})' = -\sin{x} \\ (\sin{x})' = \cos{x} \end{array}$$
6. Find the derivative of
$$x^{x}$$

Last edited: Jun 1, 2010
4. Jun 2, 2010

### Char. Limit

Shouldn't you include the definition of a derivative before introducing the rules for it? Just a thought...

$$\frac{d f\left(x\right)}{dx}= \text{lim}_{h\rightarrow0} \frac{f\left(x+h\right)-f\left(x\right)}{h}$$

To solve it, you first have to eliminate h from the denominator.