Answer:
5.2
Step-by-step explanation:
Since you have a linear function, asking for derivative is equivalent to asking for the slope.
The slope of y=5.2x+2.3 is 5.2 so the derivative is 5.2 .
However, if you really want to use the definition of derivative, you may.
That is, [tex]\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}[/tex].
We know [tex]f(x)=5.2x+2.3[/tex] so [tex]f(x+h)=5.2(x+h)+2.3[/tex]. All I did was replace any x in the 5.2x+2.3 with (x+h) to obtain f(x+h).
Let's plug it into our definition:
[tex]\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}[/tex]
[tex]\lim_{h \rightarrow 0} \frac{[5.2(x+h)+2.3]-[5.2x+2.3]}{h}[/tex]
Now we need to do some distributing. I see I need this distributive property both for the 5.2(x+h) and the -[5.2x+2.3].
[tex]\lim_{h \rightarrow 0} \frac{5.2x+5.2h+2.3-5.2x-2.3}{h}[/tex]
There are some like terms to combine in the numerator. The cool thing is they are opposites and when you add opposites you get 0.
[tex]\lim_{h \rightarrow 0} \frac{5.2h}{h}[/tex]
There is a common factor in the numerator and denominator. h/h=1.
[tex]\lim_{h \rightarrow 0}5.2[/tex]
5.2
