differentiate the following function.\n\n$y = \\frac{5x + 7 - 3e^{x}}{2}$\n\n$\\frac{dy}{dx} = \\square$

Explanation:

Step1: Simplify the function

$$y=\frac{5x}{2}+\frac{7}{2}-\frac{3e^{x}}{2}$$

Step2: Differentiate term - by - term

• For the term $\frac{5x}{2}$: Using the power rule $\frac{d}{dx}(ax^{n}) = anx^{n - 1}$ (here $n = 1$ and $a=\frac{5}{2}$), we have $\frac{d}{dx}(\frac{5x}{2})=\frac{5}{2}\times1\times x^{0}=\frac{5}{2}$.
• For the term $\frac{7}{2}$: Since the derivative of a constant $C$ (here $C = \frac{7}{2}$) is $0$, $\frac{d}{dx}(\frac{7}{2})=0$.
• For the term $-\frac{3e^{x}}{2}$: Using the rule $\frac{d}{dx}(e^{x})=e^{x}$, we get $\frac{d}{dx}(-\frac{3e^{x}}{2})=-\frac{3}{2}e^{x}$.

Step3: Combine the derivatives

$$\frac{dy}{dx}=\frac{5}{2}+0-\frac{3e^{x}}{2}=\frac{5 - 3e^{x}}{2}$$

Answer:

$\frac{5 - 3e^{x}}{2}$