Final answer :
The derivative of the function g(x) = 2x² - 1 using the difference quotient method is g'(x) = 4x.
Step-by-step explanation:
To find the derivative of the function g(x) = 2x² - 1 using the difference quotient method, we need to compute the following limit:
lim(h→0) [g(x + h) - g(x)] / h
Let's compute the difference quotient step by step:
g(x + h) = 2(x + h)² - 1
= 2(x² + 2hx + h²) - 1
= 2x² + 4hx + 2h² - 1
Substituting this back into the difference quotient:
lim(h→0) [(2x² + 4hx + 2h² - 1) - (2x² - 1)] / h
= lim(h→0) [4hx + 2h²] / h
= lim(h→0) 4x + 2h
Now, we can take the limit as h approaches 0:
lim(h→0) 4x + 2h = 4x
Therefore, the derivative of g(x) = 2x² - 1 using the difference quotient method is g'(x) = 4x.
In summary, by using the difference quotient method, we found that the derivative of the function g(x) = 2x² - 1 is g'(x) = 4x. This means that the rate at which the function changes at any given point is equal to four times the x-value. To verify this result, we can also find the derivative directly using the power rule for derivatives, taking the derivative of each term of the function g(x). However, the difference quotient method provides a more general approach to finding derivatives and is particularly useful when dealing with more complicated functions. In this case, the calculation involved expanding the squared term and simplifying the difference quotient expression before taking the limit as h approaches zero. Ultimately, we arrive at the derivative g'(x) = 4x.