Find a formula for the derivative of the function g(x)=4x2−1
using difference quotients:
g′(x)=lim[(
h→0
(g(x+h)-g(x))/h]
g′(x)= lim[(4(x+h)^2-1-4x^2+1)/h]
g′(x)= lim[(4x^2+8hx+4h^2-1-4x^2+1)/h]
g′(x)= lim[(8hx+4h^2)/h]
g′(x)= lim[8x+4h] as h→0
g′(x) = 8x
To find the derivative of the function g(x) = 4x^2 - 1 using difference quotients, we'll start by using the definition of the derivative:
g′(x) = lim[(h→0) (g(x+h) - g(x))/h]
Step 1: Substitute the function g(x) into the difference quotient formula:
g′(x) = lim[(h→0) ((4(x+h)^2 - 1) - (4x^2 - 1))/h]
Step 2: Expand and simplify the expression:
g′(x) = lim[(h→0) (4(x^2 + 2hx + h^2) - 4x^2)/h]
Step 3: Distribute the 4:
g′(x) = lim[(h→0) (4x^2 + 8hx + 4h^2 - 4x^2)/h]
Step 4: Cancel out the common terms:
g′(x) = lim[(h→0) (8hx + 4h^2)/h]
Step 5: Divide each term by h:
g′(x) = lim[(h→0) (8x + 4h)]
Step 6: Take the limit as h approaches 0:
g′(x) = 8x
Therefore, the derivative of the function g(x) = 4x^2 - 1 is g′(x) = 8x.
To find the derivative of the function g(x) = 4x^2 - 1 using difference quotients, you can use the definition of the derivative:
g'(x) = lim[h→0] [(g(x + h) - g(x)) / h]
Now, let's substitute the function g(x) into this expression:
g'(x) = lim[h→0] [(4(x + h)^2 - 1 - (4x^2 - 1)) / h]
Expanding and simplifying the expression, we get:
g'(x) = lim[h→0] [(4(x^2 + 2hx + h^2) - 1 - 4x^2 + 1) / h]
= lim[h→0] [(4x^2 + 8hx + 4h^2 - 1 - 4x^2 + 1) / h]
= lim[h→0] [(8hx + 4h^2) / h]
Now, we can simplify further by canceling out the h terms:
g'(x) = lim[h→0] [8x + 4h]
Finally, if we take the limit as h approaches 0, the h term disappears:
g'(x) = 8x
Therefore, the derivative of the function g(x) = 4x^2 - 1 is g'(x) = 8x.