Final answer:
To find the slope of the tangent line for the function f(x) = 4x² - 7x at any point, differentiate the function to get f'(x) = 8x -7, which gives the slope of the tangent line at a particular x value.
Step-by-step explanation:
To find the slope of the tangent line to the function f(x) = 4x² - 7x at any point, we will use the four-step process for finding derivatives which involves finding the derivative of the function since the derivative at a point gives us the slope of the tangent line at that point.
Step 1: Differentiate the function with respect to x.
We apply the power rule to the function f(x). The power rule states that the derivative of x raised to any power n is n times x raised to the (n-1) power.
The derivative of 4x² is 8x and the derivative of -7x is -7. Therefore, the derivative of the function, denoted as f'(x), is 8x - 7.
Step 2: Simplify the derivative.
The derivative is already simplified in this case: f'(x) = 8x - 7.
Step 3: Evaluate the derivative at a specific point.
If you are given a specific value of x, substitute that value into the derivative to find the slope of the tangent line at that point.
Step 4: Write the equation of the tangent line.
Using the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point, you can write the equation of the tangent line.