Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the given function.
\[
f(x)=3 x^{2}-7 x+4
\]
\[
\frac{f(x+h)-f(x)}{h}=\square(\text { Simplify your answer. })
\]
Answer
Final Answer: The simplified difference quotient for the function \(f(x)=3 x^{2}-7 x+4\) is \(\boxed{6x+3h-7}\)
Step 1 ：Given the function \(f(x)=3 x^{2}-7 x+4\)
Step 2 ：Substitute \(x+h\) into the function to get \(f(x+h)=3(x+h)^{2}-7(x+h)+4\)
Step 3 ：Expand \(f(x+h)\) to get \(f(x+h)=3x^{2}+6hx+3h^{2}-7x-7h+4\)
Step 4 ：Subtract \(f(x)\) from \(f(x+h)\) to get \(f(x+h)-f(x)=3x^{2}+6hx+3h^{2}-7x-7h+4-(3x^{2}-7x+4)\)
Step 5 ：Simplify the above expression to get \(f(x+h)-f(x)=6hx+3h^{2}-7h\)
Step 6 ：Divide the above expression by \(h\) to get the difference quotient \(\frac{f(x+h)-f(x)}{h}=6x+3h-7\)
Step 7 ：Final Answer: The simplified difference quotient for the function \(f(x)=3 x^{2}-7 x+4\) is \(\boxed{6x+3h-7}\)
