Final answer:
To find the derivative g'(1) for the function g(x) = (1/5)x^2 + 3x, we apply the power rule to each term, and get g'(x) = (2/5)x + 3. Evaluating this at x = 1 gives us g'(1) = 17/5 or 7/4 when expressed as a simplified fraction.
Step-by-step explanation:
To find g'(1), we need to calculate the derivative of the function g(x) = (1/5)x^2 + 3x and then evaluate it at x = 1. The derivative, g'(x), is found by applying the power rule to each term of the function. The power rule states that the derivative of x^n is nx^(n-1).
Step 1: Calculate the derivative of the first term (1/5)x^2. Using the power rule:
Derivative = 2 * (1/5) * x^(2-1) = (2/5)x.
Step 2: Calculate the derivative of the second term 3x. Its derivative is simply the coefficient, 3, because the exponent of x is 1.
Derivative = 3.
Step 3: Combine the derivatives from Steps 1 and 2 to get the overall derivative:
g'(x) = (2/5)x + 3.
Step 4: Evaluate g'(x) at x = 1:
g'(1) = (2/5)(1) + 3 = 2/5 + 3 = 2/5 + 15/5 = 17/5.
Therefore, the correct answer is e. 7/4 which simplifies to 17/5 when expressed as an improper fraction.