Every point of a curve has a tangent, and the tangent has a gradient. [br]Recall for y = mx + c the line has a gradient of m.[br]Lets investigate the gradient function, the new function we get from the gradient of curve f.[br][br]1. Drag the slider [b]a[/b] to see what happens to the tangent to the curve at the point (a,f(a))[br]2. Select [b]show gradient value[/b] to see a blue point which shows (a, f'(a)) - the x value is a, the y value is the gradient (or slope) of the tangent [br]3. [b]"set trace"[/b] and now the blue points remain as you move the slider a.[br]4. Can you see a pattern in the trace points?[br] This line or curve shows the gradient of the tangent to the curve y=f(x) for all values of x and is written f'(x) (f prime of x, sometimes called f dash). We can call it the gradient function.[br]5. what is f'(x)? have a guess - say f'(x)=4x. Type your guess into the [color=#ff0000]red input box[/color] and see if you were right. (you can hide/show the guess)[br]When you have got the fright function for f'(x), make a note and move on to the next f(x) function (see below)[br][br]6. Try another function for f(x) by typing, say x^3 in the box.[br]

[i]This is an investigation, so look for patterns and try to predict what the next answer will be.[br][/i][br]1. Find f'(x) when [br]a) f(x) = [math]x^2[/math][br]b) f(x) = [math]x^3[/math][br]c) f(x) = [math]x^4[/math][br]d) guess for f(x) = [math]x^n[/math] (and predict for n=7)[br][br]2. Find f'(x) when [br]a) f(x) = [math]4x[/math][br]b) f(x) = [math]x^2+3x[/math][br]c) f(x) = [math]-3x^2+4x+5[/math][br][br][br]3. [b]Advanced [/b]Find f'(x) when [br]a) f(x) = 1/x[br]b) f(x) = 1/x^2[br][br]c) f(x) = sin(x)[br]d) f(x) = cos(x)[br]e) f(x) = [math]e^x[/math][br]f) guess for f(x) = ln(x)]