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.


a. 6ƒ(x) - g(x), x = 1

Verified step by step guidance

1. To find the derivative of the combination 6ƒ(x) - g(x), we need to apply the rules of differentiation. Specifically, we will use the constant multiple rule and the sum/difference rule.
2. The constant multiple rule states that if you have a constant multiplied by a function, the derivative is the constant multiplied by the derivative of the function. So, the derivative of 6ƒ(x) is 6ƒ'(x).
3. The sum/difference rule states that the derivative of a sum or difference of functions is the sum or difference of their derivatives. Therefore, the derivative of 6ƒ(x) - g(x) is 6ƒ'(x) - g'(x).
4. Now, substitute the given values for the derivatives at x = 1. We have ƒ'(1) = 1/2 and g'(1) = -4.
5. Finally, calculate the expression 6ƒ'(1) - g'(1) using the substituted values: 6 * (1/2) - (-4).

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 calculating the derivatives of their combinations.

Linear Combinations of Functions

A linear combination of functions involves adding or subtracting functions multiplied by constants. For example, in the expression 6ƒ(x) - g(x), the function ƒ(x) is scaled by 6 and then g(x) is subtracted. Understanding how to differentiate such combinations using the properties of derivatives is essential for solving the problem.

Application of the Chain Rule and Sum Rule

The chain rule and sum rule are fundamental rules in calculus for finding derivatives. The sum rule states that the derivative of a sum of functions is the sum of their derivatives, while the chain rule is used when differentiating composite functions. In this question, applying these rules will allow us to find the derivative of the combination 6ƒ(x) - g(x) at x = 1.