# 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 $$\frac{1}{x}$$.