2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

The derivative rules for constants, sums, differences, and constant multiples are fundamental concepts in Calculus that allow us to find the derivative of more complex functions by breaking them down into simpler terms. The Constant Rule states that the derivative of a constant is always zero. This means that if we have a function of the form f(x) = c, where c is a constant, the derivative of the function is 0. For example, the derivative of f(x) = 3 is 0. The Sum Rule states that the derivative of a sum of functions is equal to the sum of the derivatives of each function. This means that if we have a function of the form f(x) = g(x) + h(x), the derivative of the function is f'(x) = g'(x) + h'(x). For example, the derivative of f(x) = 2x + 3 is 2. The Difference Rule states that the derivative of a difference of functions is equal to the difference of the derivatives of each function. This means that if we have a function of the form f(x) = g(x) - h(x), the derivative of the function is f'(x) = g'(x) - h'(x). For example, the derivative of f(x) = 2x - 3 is 2. The Constant Multiple Rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. This means that if we have a function of the form f(x) = cg(x), where c is a constant, the derivative of the function is f'(x) = cg'(x). For example, the derivative of f(x) = 3x is 3. It is important to note that these rules are only applicable when the exponent of x is a constant. If the exponent is a variable or a function, different methods such as the chain rule must be used to find the derivative.

Example 1: Find the derivative of f(x) = 4x + 2 Solution: Using the Sum Rule, we know that the derivative of a sum of functions is equal to the sum of the derivatives of each function. In this case, the function can be written as f(x) = 4x + 2. Using the constant rule, the derivative of the constant 2 is 0. The derivative of 4x is 4. So, the derivative of f(x) = 4x + 2 is 4 + 0 = 4. Example 2: Find the derivative of f(x) = 3x - 5 Solution: Using the Difference Rule, we know that the derivative of a difference of functions is equal to the difference of the derivatives of each function. In this case, the function can be written as f(x) = 3x - 5. The derivative of 3x is 3. Using the constant rule, the derivative of the constant -5 is 0. So, the derivative of f(x) = 3x - 5 is 3 - 0 = 3 Example 3: Find the derivative of f(x) = 5(2x+3) Solution: Using the Constant Multiple Rule, we know that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. In this case, the function can be written as f(x) = 5(2x+3) . The derivative of 2x+3 is 2. So, the derivative of f(x) = 5(2x+3) is 5*2 = 10

Example 4:

Find the derivative of f(x) = 2x^2 + 3x + 4

Solution: Using the Sum Rule, we know that the derivative of a sum of functions is equal to the sum of the derivatives of each function. In this case, the function can be written as f(x) = 2x^2 + 3x + 4. Using the power rule, the derivative of 2x^2 is 4x. The derivative of 3x is 3. Using the constant rule, the derivative of the constant 4 is 0. So, the derivative of f(x) = 2x^2 + 3x + 4 is 4x + 3 + 0 = 4x + 3

Example 5:

Find the derivative of f(x) = 4x^3 - 2x^2 + 3x - 1

Solution: Using the Sum Rule, we know that the derivative of a sum of functions is equal to the sum of the derivatives of each function. In this case, the function can be written as f(x) = 4x^3 - 2x^2 + 3x - 1. Using the power rule, the derivative of 4x^3 is 12x^2, the derivative of -2x^2 is -4x, the derivative of 3x is 3, and the derivative of -1 is 0. So, the derivative of f(x) = 4x^3 - 2x^2 + 3x - 1 is 12x^2 - 4x + 3 + 0 = 12x^2 - 4x + 3

It is important to remember that these rules are only a starting point and that as the complexity of functions increases, more advanced techniques such as the chain rule may be needed to find the derivative. However, understanding and applying these basic rules can greatly simplify the process of finding derivatives.