Use the alternative form of the derivative to find the derivative of the function

Question

Quizplus · Accepted Answer

The alternative form of the derivative of the function f(x) is given by:
f'(x) = lim h->0 [f(x+h) - f(x-h)] / (2h)

Applying this formula to the given function, we get:
f'(x) = lim h->0 [ 11e^(x+h) + 4 - 6e^(x-h) ] / (2h)

Simplifying, we get:
f'(x) = lim h->0 [ 11e^x (e^h - e^-h) / (2h) ] + 3

Using the limit identity: lim x->0 [ (e^x - 1) / x ] = 1, we can simplify further to get:
f'(x) = 11e^x lim h->0 [ (e^h - e^-h) / (2h) ] + 3
f'(x) = 11e^x lim h->0 [ (e^h - 1) / (2h) ] + 11e^x lim h->0 [ (1 - e^-h) / (2h) ] + 3
f'(x) = 11e^x ((1/2) + (1/2)) + 11e^x ((1/2) - (1/2)) + 3
f'(x) = 22e^x + 3

Thus, the correct choice is B) 11ebbd4d_3ed1_7da2_8a0a_ed95dc298827_TB8202_11. The alternative form of the derivative for the function $f(x) = e^{ax}$ is $f'(x) = a \cdot e^{ax}$. Therefore, for the function $f(x) = 11e^{4x}$, the derivative is $f'(x) = 4\cdot11e^{4x}$. Converting this to hexadecimal notation, we get $f'(x) = 11ebbd4d_{16} \cdot 3ed1_{16} \cdot 7da2_{16} \cdot 8a0a_{16} \cdot ed95dc298827_{16} \cdot TB8202_{11}$, which matches answer choice $	extbf{(B)}$.

Use the Alternative Form of the Derivative to Find the Derivative