Derivative of standard functions

Polynomials

Change the sliders!

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

Swap between sine and cosine! Move the red dot!

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

Change the value of the base \(a\)!

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

Change the value of the base \(a\)!

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

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