Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. $$ g(x)=x^{2}+5 x $$

Step-by-Step Solution

Answer

The derivative of the function \( g(x) = x^2 + 5x \) is \( g'(x) = 2x + 5 \).

Step 1: Identify Functions and Constants

The function given is \( g(x) = x^{2} + 5x \). You can see that this is composed of two parts: \( x^2 \) (a function with power 2) and \( 5x \) (a constant 5 times a function).

Step 2: Apply the Power Rule

Start with the function \( x^2 \). By applying the power rule, where \( (x^n)' = n\cdot x^{(n-1)} \), the derivative of the function \( x^2 \) is \( 2\cdot x^{(2-1)} \). So, the derivative of \( x^2 \) is \( 2x \).

Step 3: Apply the Constant Rule

Next, apply the constant rule to the function \( 5x \). According to the constant rule, the derivative of a constant times a function is just the constant times the derivative of the function. Thus, the derivative of \( 5x \) will be \( 5 \cdot 1 \) since the derivative of \( x \) is \( 1 \). This gives you \( 5 \).

Step 4: Combine the Results

Lastly, combine the derivatives from the previous steps. This gives \( 2x + 5 \) as the final result.