Find the derivative of the function by the limit process.

Question

Find the derivative of the function  by the limit process.

Quizplus · Accepted Answer

We can use the limit definition of the derivative:
f'(x) = lim (h->0) [f(x+h) - f(x)]/h
Plugging in the given function, we get:
f'(x) = lim (h->0) [(11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x+h))^2 - (11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x))^2]/h
Using the difference of squares, we can simplify this expression to:
f'(x) = lim (h->0) [11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x+h) + 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x)][11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x+h) - 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x)]/h
Canceling out the common factor of [11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x+h) + 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x)], we get:
f'(x) = lim (h->0) [11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x+h) - 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x)]/h + lim (h->0) [11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x+h) - 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x)]
The first limit is the definition of the derivative of 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x), so it is equal to 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11'(x). The second limit is just the difference quotient of 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x), evaluated at x and x+h. Taking the limit as h approaches 0, this becomes the derivative of 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x) again. Therefore, we have:
f'(x) = 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11'(x) + 11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11'(x)
f'(x) = 2*11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11'(x)
Substituting in the given function, we get:
f'(x) = 2*(11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x))*(11ec1854_fdbb_9058_acfc_d5ee21abf396_TBX8702_11 - 11ec1854_fdbb_9059_acfc_d7793c317cd7_TBX8702_11)
Simplifying this expression, we get:
f'(x) = 2*(11ec1854_fdbb_6945_acfc_c303a6d74000_TBX8702_11(x))*(11ec1854_fdbb_905a_acfc_f330fde59598_TBX8702_11) = 11ec1854_fdbb_9058_acfc_d5ee21abf396_TBX8702_11

Find the Derivative of the Function by the Limit