How do you find the slope of the graph #f(x)=(2x+1)^2# at (0,1)?

Question

Charles Anctil · Accepted Answer

Using the chain rule of derivatives, and then plugging in a x value of 0 for the #f'(x)# function, you get a slope.

Consider #F(x)=fprime(g(x))*gprime(x) # This what we define as the chain rule, "first the outside derivative then the inside derivative" If #f(x)=(2x+1)^2# and #f'(x)=8x+4# and when #f'(0)=8(0)+4=4# Then the slope equals 4. Proof Thing: Derivatives are slopes of functions. We know this by the formal definition of #f'(x)=(f(x+h)-f(x))/h# and #m=(y_2-y_1)/(x_2-x_1)# They are similar because that is how the definition came by substituting them as #(Deltay)/(Deltax)#; where #h# represents the change the function. So, derivatives find a slope of a function. We simplify the process by using various derivative rules

Cameron Alexander · Answer

undefined To find the slope of the graph of $ f(x) = (2x + 1)^2 $ at the point (0,1), you need to calculate the derivative of the function and then evaluate it at $ x = 0 $.

First, find the derivative of $ f(x) $:
$$ f'(x) = 2(2x + 1)(2) = 8x + 4 $$

Next, evaluate the derivative at $ x = 0 $:
$$ f'(0) = 8(0) + 4 = 4 $$

So, the slope of the graph of $ f(x) $ at the point (0,1) is $ 4 $.