All About the Derivative Function
f(x)= x^2 -6x +10
Delta y/ Delta x
(2-5)/(2-1) = -3
(5-10)/(1-0) = -5
((-3)+(-5))/2 = -4
Another way of finding the derivative is by using the definition of the derivative with algebra to solve for d.
First off, evaluate using the function f(x) = x^2 -6x +10 for f(1) and for f(1+h).
f(1) = (1)^2 -6(1) +10
f(1) = 5
f(1+h) = (1+h)^2 -6(1+h) +10
f(1+h) = 1+ 2h +h^2 -6 -6h +10
f(1+h) = h^2 -4h +5
lim (f(1+h)- f(1))/h
lim ((h^2 -4h +5) -(5))/h
lim (h^2 -4h)/h
Assume h = 0, therefore d = -4.
You can determine a function that will give you every derivative value for the function you are using. You use the formal definition of a derivative to solve for f^1 (x).
f(x) = x^2 -6x +10
f(x+h) = (x+h)^2 -6(x+h)+10
f(x+h) = x^2 +2xh +h^2 -6x -6h +10
lim ((x^2 -6x +10 +h^2 + 2xh -6h)-(x^2 -6x +10))/h
lim (h(h +2x -6))/h
lim h +2x -6
f^1(x) = 2x -6
You can also draw a tangent line if you are given a graph that is draw to scale. Your ruler can be important if you have a graph.
Your calculator has a SLOPE program and you can use this program to calculate the derivative but it is still only an estimate. You can also use your calculator to draw a tangent line on the graph of the function.
In case you forget what the formal definition of a derivative is:
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