91Ó°ÊÓ

The constant multiple rule is very useful when differentiating functions that are scaled by a constant factor. The rule states that if you have a constant multiplied by a function, you can take the constant outside the derivative. For example, if you have a function of the form \[y = c \times f(x)\]where `c` is a constant, then its derivative is simply:\[ \frac{d}{dx} [c \times f(x)] = c \times \frac{d}{dx}[f(x)] \]In our specific problem, the function was given as:\[ y = \frac{1}{4} (2x + 4 - 5e^x) \]By applying the constant multiple rule, the differentiation process becomes simpler, allowing us to focus on differentiating the individual terms inside the parentheses. This makes it straightforward to manage the constants and streamline the differentiation process.

Exponential functions are an important class of functions in calculus, denoted as \( e^x \), where `e` is Euler's number (approximately 2.718). When differentiating exponential functions, one key property stands out: the derivative of \( e^x \) is \( e^x \). Here's the differentiation rule for exponential functions:\[ \frac{d}{dx} [e^x] = e^x \]In our problem, we encounter the term \( -5e^x \). Differentiating it involves the constant multiple rule mentioned earlier:\[ \frac{d}{dx} [-5e^x] = -5 \times \frac{d}{dx} [e^x] = -5e^x \]This property of exponential functions allows for straightforward differentiation and is essential in many calculus problems.

Polynomials are expressions involving variables raised to whole number powers. Differentiating polynomials is generally easy and follows a simple rule: If you have a term like \( ax^n \), where `a` is a constant and `n` is a non-negative integer, its derivative is given by:\[ \frac{d}{dx} [ax^n] = a \times n \times x^{n-1} \]For example, the derivative of \( 2x \) is:\[ \frac{d}{dx} [2x] = 2 \]and the derivative of a constant, such as \( 4 \), is zero.In our exercise, the function \( y = \frac{1}{4} (2x + 4 - 5e^x) \) contains a polynomial part \( 2x + 4 \). Applying the rules above, we find that:\[ \frac{d}{dx}[2x] = 2 \]and\[ \frac{d}{dx}[4] = 0 \]Combining these individual derivatives helps us build the final result efficiently.