Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of \(\sqrt{x}\). $$g(w)=\left\\{\begin{array}{ll} w+5 e^{w} & \text { if } w \leq 1 \\ 2 w^{3}+4 w+5 & \text { if } w>1 \end{array}\right.$$

Short answer

The derivative of the given function \(g(w)\) is: $$ g'(w)=\left\\{\begin{array}{ll} 1 + 5 e^{w} & \text { if } w \leq 1 \\\ 6w^2 + 4 & \text { if } w>1 \end{array}\right.$$

Step-by-step solution

Find the derivative for the first case (\(w \leq 1\))

The function for this case is \(w + 5e^w\). Now apply differentiation rules: $$ \frac{d}{dw}(w) = 1$$ $$ \frac{d}{dw}(5 e^w) = 5\frac{d}{dw}(e^{w}) = 5e^w$$ Now add the derivatives: $$ \frac{d}{dw}(w + 5e^w) = 1 + 5e^w $$

Find the derivative for the second case (\(w > 1\))

The function for this case is \(2 w^{3}+4 w+5\). We apply differentiation rules as follows: $$ \frac{d}{dw}(2 w^{3}) = 6w^2 $$ $$ \frac{d}{dw}(4 w) = 4 $$ $$ \frac{d}{dw}(5) = 0 $$ Now add the derivatives: $$ \frac{d}{dw}(2 w^{3} + 4 w + 5) = 6w^2 + 4 $$

Combine the derivatives

Now that we have found the derivatives for both cases, we will combine them into a piecewise function: $$ g'(w)=\left\\{ \begin{array}{ll} 1 + 5e^w & \text { if } w \leq 1 \\\ 6w^2 + 4 & \text { if } w>1 \end{array}\right.$$ The derivative of the given function \(g(w)\) is: $$ g'(w)=\left\\{\begin{array}{ll} 1 + 5 e^{w} & \text { if } w \leq 1 \\\ 6w^2 + 4 & \text { if } w>1 \end{array}\right.$$

Key concepts

Differentiation Rules

Differentiation is a key concept in calculus, and it involves finding the rate at which a function is changing at any given point. The rules for differentiation help simplify this process for various types of functions. Here are a few basic differentiation rules applied in our exercise to find derivatives of piecewise functions:

• Power Rule: This rule states that if you have a function in the form of \(x^n\), then its derivative is \(nx^{n-1}\). For example, the derivative of \(2w^3\) is \(6w^2\).
• Constant Rule: The derivative of a constant (a number on its own) is always zero. For instance, the derivative of 5 is zero.
• Sum Rule: If you have a function that is the sum of two or more functions, the derivative of the total function is the sum of each of the individual derivatives.

Using these rules, we can differentiate piecewise functions by applying the appropriate rules to each piece.

Exponential Function Derivative

The exponential function is unique because its derivative is itself. The general form of the exponential function is \(e^x\), and its derivative is also \(e^x\). In our exercise, we had to differentiate the function part \(5e^w\). Here's how it works:

• Basic Exponential Derivative: If you have \(e^x\), the derivative is simply \(e^x\).
• Constant Multiple Rule: When the exponential function is multiplied by a constant, you multiply the derivative by the same constant. So, the derivative of \(5e^w\) becomes \(5e^w\).

These concepts highlight how efficiently exponential functions can be differentiated, as they retain their form through differentiation.

Polynomial Function Derivative

Polynomial functions are made up of terms which are powers of a variable, like \(w^3\) or \(w\), often with coefficients. To find the derivative of polynomial functions like \(2w^3 + 4w + 5\), we apply the power and constant rules. Here's the process:

• Power Rule Application: For each term that is a power of the variable, apply the power rule. This makes calculating each part straightforward, such as turning \(2w^3\) into \(6w^2\).
• Constant Term Handling: Remember, the derivative of a constant is zero, so any standalone number without a variable, like \(5\), disappears in differentiation.
• Adding Up Derivatives: Once you've differentiated each part separately, simply add them together to get the full derivative for the polynomial function.

Differentiating polynomial functions becomes easy by applying these principles, breaking down complex expressions into simple, manageable pieces.