compute the following derivatives using the formulas below.\n\n$m = \\frac { d } { d x } ( e ^ { x } ) | _ {…

compute the following derivatives using the formulas below.\n\n$m = \\frac { d } { d x } ( e ^ { x } ) | _ { x = 0 } = 1$\n\n$\\frac { d } { d x } e ^ { x } = m e ^ { x }$\n\n(a) $\\frac { d } { d x } ( e ^ { x } ) | _ { x = 1 }$\n\n(b) $\\frac { d } { d x } ( e ^ { x } ) | _ { x = - 2 }$\n

compute the following derivatives using the formulas below.\n\n$m = \\frac { d } { d x } ( e ^ { x } ) | _ { x = 0 } = 1$\n\n$\\frac { d } { d x } e ^ { x } = m e ^ { x }$\n\n(a) $\\frac { d } { d x } ( e ^ { x } ) | _ { x = 1 }$\n\n(b) $\\frac { d } { d x } ( e ^ { x } ) | _ { x = - 2 }$\n

Answer

Explanation:

Step1: Recall the derivative formula

Given (\frac{d}{dx}e^{x}=me^{x}) and (m = 1), so (\frac{d}{dx}e^{x}=e^{x}).

Step2: Evaluate the derivative at (x = 1) for part (a)

Substitute (x = 1) into (\frac{d}{dx}e^{x}=e^{x}). We get (\left.\frac{d}{dx}(e^{x})\right|_{x = 1}=e^{1}=e).

Step3: Evaluate the derivative at (x=-2) for part (b)

Substitute (x=-2) into (\frac{d}{dx}e^{x}=e^{x}). We get (\left.\frac{d}{dx}(e^{x})\right|_{x=-2}=e^{-2}=\frac{1}{e^{2}}).

Answer:

(a) (e) (b) (\frac{1}{e^{2}})