Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


c. ƒ(x) , x = 1
g(x) + 1

Verified step by step guidance

1. Identify the function combination for which you need to find the derivative. In this case, it is the derivative of the function f(x) with respect to x, evaluated at x = 1.
2. Recall that the derivative of a function at a point gives the rate of change of the function at that point. Here, you need to find f'(x) at x = 1.
3. From the given data, locate the value of f'(x) at x = 1. According to the table, f'(1) = 1/2.
4. Since the problem asks for the derivative of f(x) at x = 1, and you have already identified f'(1) = 1/2, this is the value you need.
5. Thus, the first derivative of f(x) at x = 1 is 1/2, which represents the slope of the tangent line to the curve of f(x) at that point.

Key Concepts

Derivative of a Function

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, knowing the derivatives of functions ƒ(x) and g(x) at specific points is crucial for finding the derivatives of their combinations.

Sum Rule of Derivatives

The sum rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if h(x) = ƒ(x) + g(x), then h'(x) = ƒ'(x) + g'(x). This rule is essential for solving the given problem, as it allows us to find the derivative of the combination ƒ(x) + 1 by simply using the derivative of ƒ(x) since the derivative of a constant (1) is zero.

Evaluating Derivatives at Specific Points

Evaluating a derivative at a specific point involves substituting the value of x into the derivative function. In this case, we need to find the derivative of the combination at x = 1. This requires using the provided values of the derivatives at that point to compute the final result accurately.