# Derivative of standard functions

## Polynomials

### Exercises - guess the derivative function!

#### The constant function

Let \(a=b=c=d=0\). Find a formula for \(f'(x)\)! Explain!

#### First degree polynomials

Let \(a=b=c=0\), try different values of \(d\). Find an **exact**
formula for \(f'(x)\)! Explain!

What happens with the graph of \(f'(x)\) when you change the slider
`e`

? Explain!

#### Higher degree polynomials

What kind of a function is \(f'(x)\) when \(f(x)\) is a second degree polynomial?

What kind of a function is \(f'(x)\) when \(f(x)\) is a third degree polynomial? Make a guess!

What kind of a function is \(f'(x)\) when \(f(x)\) is a fourth degree polynomial? Make a guess!

What happens with the graph of \(f'(x)\) when you change the slider
`e`

?

#### The derivative of a polynomial of degree n

Try to find an **exact** formula for \(f'(x)\)
when \(f(x)=cx^2\)
(let the sliders \(a, b, d, e\) be zero). How is the gradient of the
line related to the number \(c\)? Try to figure it out by changing the
slider `c`.

Having the formulae for \(f'(x)\) when \(f(x)=ax^0, f(x)=ax^1, f(x)=ax^2\); try to guess the formula for \(f'(x)\) when \(f(x)=ax^3\) and \(f(x)=ax^4\).

#### Antiderivative

If \(f'(x)=3\) what is \(f(x)\)? Could many functions have the same derivative functions?

If \(f'(x)=4x\) what is \(f(x)\)?

If \(f'(x)=6x^2\) what is \(f(x)\)?

If \(f'(x)=x^{-1}\) what is \(f(x)\)?

## Sine and cosine - guess the derivative

The gray point has the slope of the tangent line as its `y`-value. Drag the red point
to see the trace of the gray point.

Guess the derivative function of \(f(x)=\sin (ax)\) when \(a=1\).

Guess the derivative function of \(f(x)=\sin (ax)\) when \(a\neq 1\).

Change to the function \(f(x)=\cos (ax)\) by changing the left slider. Guess the derivative function of \(f(x)=\cos (ax)\) when \(a=1\).

Guess the derivative function of \(f(x)=\cos (ax)\) when \(a\neq 1\).

## Exponential function - guess the base

To the left in the applet above, the graph of the function \(f(x)=a^x\) and its derivative function is shown, to right the difference between these two functions is shown.

Change the base of the exponential function by entering a new value of \(a\) in the input box.

Find a value of the base that makes the two functions as close as possible, the value should be correct to five significant figures.

## Derivative of the logarithmic function using a convenient base

Try to find a base \(a\) such that the derivative is \(\frac{1}{x}\).

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License