To solve the problem of evaluating and simplifying the expression [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] for the function [tex]\(f(x) = 6x^2 + x\)[/tex], follow these steps:
### Step 1: Calculate [tex]\(f(x + h)\)[/tex]
First, we need to find the value of the function at [tex]\(x + h\)[/tex]. This involves substituting [tex]\(x + h\)[/tex] into the function [tex]\(f(x)\)[/tex].
[tex]\[ f(x + h) = 6(x + h)^2 + (x + h) \][/tex]
### Step 2: Expand [tex]\(f(x + h)\)[/tex]
Next, expand the expression for [tex]\(f(x + h)\)[/tex]:
[tex]\[ 6(x + h)^2 + (x + h) \][/tex]
[tex]\[ = 6(x^2 + 2xh + h^2) + x + h \][/tex]
[tex]\[ = 6x^2 + 12xh + 6h^2 + x + h \][/tex]
So,
[tex]\[ f(x + h) = 6x^2 + 12xh + 6h^2 + x + h \][/tex]
### Step 3: Calculate [tex]\(f(x + h) - f(x)\)[/tex]
Now, subtract [tex]\(f(x)\)[/tex] from [tex]\(f(x + h)\)[/tex]:
[tex]\[ f(x + h) - f(x) = (6x^2 + 12xh + 6h^2 + x + h) - (6x^2 + x) \][/tex]
[tex]\[ = 6x^2 + 12xh + 6h^2 + x + h - 6x^2 - x \][/tex]
Combining like terms, we get:
[tex]\[ f(x + h) - f(x) = 12xh + 6h^2 + h \][/tex]
### Step 4: Divide by [tex]\(h\)[/tex]
Next, divide the result by [tex]\(h\)[/tex]:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{12xh + 6h^2 + h}{h} \][/tex]
### Step 5: Simplify the expression
Factor out an [tex]\(h\)[/tex] from the numerator:
[tex]\[ \frac{12xh + 6h^2 + h}{h} = \frac{h(12x + 6h + 1)}{h} \][/tex]
Cancel the [tex]\(h\)[/tex] in the numerator and denominator:
[tex]\[ \frac{f(x + h) - f(x)}{h} = 12x + 6h + 1 \][/tex]
Hence, the simplified form of the expression [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] is:
[tex]\[ \boxed{12x + 6h + 1} \][/tex]
