Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
a. d/dx (f(x)+2g(x)) |x=3

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the derivative of the function h(x) = f(x) + 2g(x) at x = 3.
2. Step 2: Use the linearity of derivatives. The derivative of a sum is the sum of the derivatives. So, d/dx (f(x) + 2g(x)) = f'(x) + 2g'(x).
3. Step 3: Evaluate the derivative at x = 3. We need to find f'(3) and g'(3) from the table.
4. Step 4: Substitute the values from the table into the expression f'(3) + 2g'(3).
5. Step 5: Simplify the expression to find the value of the derivative at x = 3.

Key Concepts

Derivative

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that quantifies how a function's output changes as its input changes. The notation d/dx indicates differentiation with respect to x, and derivatives can be interpreted as slopes of tangent lines to the graph of the function.

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 f(x) and g(x) are functions, then d/dx (f(x) + g(x)) = f'(x) + g'(x). This rule simplifies the process of finding derivatives when dealing with expressions that involve the addition of multiple functions.

Evaluating Derivatives at a Point

Evaluating a derivative at a specific point involves substituting the value of the variable into the derivative expression. For example, to find d/dx (f(x) + 2g(x)) at x=3, one must first compute the derivative and then substitute x=3 into the resulting expression. This process provides the instantaneous rate of change of the function at that particular point.