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Chapter 1: Problem 74

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=6 x+1$$

Short Answer

Expert verified
The simplified form of the difference quotient for the given function is 6.

Step by step solution

01

Substitute the Function Values

Start by substituting the function \(f(x) = 6x + 1\) into the difference quotient formula. So instead of \(f(x+h)\), we plug \(6(x+h)+1\), and instead of \(f(x)\), we use \(6x+1\). The difference quotient becomes \(\frac{6(x+h)+1-(6x+1)}{h}\).
02

Simplify the Expression

Now, simplify the expression in the numerator. The expression becomes \(\frac{6x+6h+1-6x-1}{h}\). Then, \(6x-6x\) and \(1-1\) cancel out, so we are left with \(\frac{6h}{h}\).
03

Simplify Further to Get the Result

Finally, simplify \(\frac{6h}{h}\) which gives a result of 6 - this is as \(h\) cancels out in the numerator and the denominator.

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