Questions: Find f'(x) = lim as h approaches 0 of (f(x+h)-f(x))/h for f(x) = 3x^2 + x - 7.

Transcript text: Find $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=3 x^{2}+x-7$.

Solution

Solution Steps

To find the derivative of the function $ f(x) = 3x^2 + x - 7 $ using the limit definition of a derivative, we will substitute $ f(x) $ into the formula $ f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} $. We will then simplify the expression and take the limit as $ h $ approaches 0.

Step 1: Define the Function

We start with the function $ f(x) = 3x^2 + x - 7 $.

Step 2: Apply the Limit Definition of the Derivative

The derivative of $ f(x) $ is given by the limit definition: \[ f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} \]

Step 3: Substitute and Simplify

Substitute $ f(x) $ into the limit definition: \[ f(x+h) = 3(x+h)^2 + (x+h) - 7 \] \[ = 3(x^2 + 2xh + h^2) + x + h - 7 \] \[ = 3x^2 + 6xh + 3h^2 + x + h - 7 \]

Now, calculate $ f(x+h) - f(x) $: \[ f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 + x + h - 7) - (3x^2 + x - 7) \] \[ = 6xh + 3h^2 + h \]

Substitute back into the limit expression: \[ f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{6xh + 3h^2 + h}{h} \]

Step 4: Simplify the Expression

Simplify the expression by dividing each term by $ h $: \[ = \lim_{h \rightarrow 0} (6x + 3h + 1) \]

Step 5: Evaluate the Limit

Evaluate the limit as $ h $ approaches 0: \[ f^{\prime}(x) = 6x + 1 \]