Introduction to Derivatives

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  10/17/2014 Introduction to Derivatives 1/6 Introduction to Derivatives It is all about slope! Slope  = Change in YChange in X  We can find an average  slope between two points. But how do we find the slope at a point ?There is nothing to measure! But with derivatives we use a small difference ...  10/17/2014 Introduction to Derivatives 2/6 ... then have it shrink towards zero . Let us Find a Derivative! We will use the slope formula: Slope  = Change in Y = Δ yChange in X Δ xto find the derivative of a function y = f(x) And follow these steps:ã Fill in this slopeformula:  Δy  = f(x+Δx) − f(x)ΔxΔx ã Simplify it as best we can,ã Then make Δx  shrink towards zero.Here we go: Example: the function f(x) = x 2 We know f(x) = x 2 , and can calculate f(x+Δx) :Start with: f(x+Δx) = (x+Δx) 2 Expand (x + Δx) 2 :  f(x+Δx) = x 2  + 2x Δx + (Δx) 2  10/17/2014 Introduction to Derivatives 3/6  Start with the slope formula:f(x+Δx) − f(x)ΔxPut in f(x+Δx)  and f(x) :x 2  + 2x Δx + (Δx) 2  − x 2 ΔxSimplify (x 2  and −x 2  cancel): = 2x Δx + (Δx) 2 ΔxSimplify more (divide through by Δx):  = 2x + Δx  And then as Δx heads towards 0  we get: = 2x   Result: the derivative of x 2  is 2x  We write dx  instead of Δx heads towards 0 , so the derivative of is commonlywritten x 2  = 2x The derivative of x 2  equals 2x or simply d dx of x 2  equals 2x What does x 2  = 2x mean? It means that, for the function x 2 , the slope or rate of change at any point is 2x .So when x=2  the slope is 2x = 4 , as shown here:Or when x=5  the slope is 2x = 10 , and so on.  10/17/2014 Introduction to Derivatives 4/6 Note: sometimes f’(x)  is also used for the derivative of : f’(x) = 2x The derivative of f(x)  equals 2x  Let's try another example. Example: What is x 3  ? We know f(x) = x 3 , and can calculate f(x+Δx) :Start with: f(x+Δx) = (x+Δx) 3 Expand (x + Δx) 3 :  f(x+Δx) = x 3  + 3x 2  Δx + 3x (Δx) 2  + (Δx) 3  The slope formula:f(x+Δx) − f(x) ΔxPut in f(x+Δx)  and f(x) :x 3  + 3x 2  Δx + 3x (Δx) 2  + (Δx) 3  − x 3 ΔxSimplify (x 3  and −x 3  cancel): = 3x 2  Δx + 3x (Δx) 2  + (Δx) 3 ΔxSimplify more (divide through by Δx):  = 3x 2  + 3x Δx + (Δx) 2  And then as Δx heads towards 0  weget:   x 3  = 3x 2 Have a play with it using the Derivative Plotter.
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