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Although using the true definition of the derivative actually makes taking derivatives harder, it is instructive since it teaches what the derivative is, and how it works.
Definition of the Derivative
The derivative of a function f(x) which is called f ' (x) is defined as
=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} "f\prime(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}")
This definition basically says that the derivative is the instantaneous slope at any given point along some function f(x). That is because as h goes to 0, the function approaches the value of the slope at some value x.
Let us see how we can use the definition to find the derivative of some functions
Examples involving the definition of a derivative
Use the definition of the derivative to find the derivative of the function
=2x+4 "1. \,\, f(x)=2x+4")
first lets define what f(x+h) is.
=2(x+h)+4=2x+2h+4 "f(x+h)=2(x+h)+4=2x+2h+4")
now, lets find the derivative using the definition
=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{(2x+2h+4)-(2x+4)}{h}=\lim_{h\to0}\frac{2h}{h}=\lim_{h\to0}\,2=2 "f\prime(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{(2x+2h+4)-(2x+4)}{h}=\lim_{h\to0}\frac{2h}{h}=\lim_{h\to0}\,2=2")
hence, we see that the derivative is 2. This means that the slope will be 2, for any given value of x. That makes sense since the function was defined as f(x)=2x+4, and for any linear function in the form y=mx+b, where m is the slope, we see that it is 2.
=x^2+2x "2. \,\, f(x)=x^2+2x")
first lets define what f(x+h) is.
=(x+h)^2+2(x+h) = x^2+2xh+h^2+2x+2h "f(x+h)=(x+h)^2+2(x+h) = x^2+2xh+h^2+2x+2h")
now, lets find the derivative using the definition
=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{(x^2+2xh+h^2+2x+2h)-(x^2+2x)}{h}=\lim_{h\to0}\frac{2xh+h^2+2h}{h}=\lim_{h\to0}\,2x+h+2=2x+2 "f\prime(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{(x^2+2xh+h^2+2x+2h)-(x^2+2x)}{h}=\lim_{h\to0}\frac{2xh+h^2+2h}{h}=\lim_{h\to0}\,2x+h+2=2x+2")
therefore, the derivative is f ' (x) = 2x+2. That means we can find the slope at any given value of x for the function. For instance, if we wanted to know what the slope of the function is at x=0, then f ' (0) = 2(0)+2 = 2. The slope is 2, when x = 0. The slope will vary depending at what point of the graph you are since it is a parabola.
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