Derivative of the Sum of Two Functions

Oct 13, 2025 by miodeya.com 46 views

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. When dealing with composite functions, we can use the chain rule to find the derivative of the composite function. However, when dealing with functions that involve addition, subtraction, multiplication, and division, we need to use the sum rule, product rule, and quotient rule to find the derivative.

Given Values

We are given the following values:

$f(3) = 4$
$g(3) = 2$
$f^{\prime}(3) = -5$
$g^{\prime}(3) = 6$

Derivative of the Sum of Two Functions

The derivative of the sum of two functions is given by the sum rule:

(f+g)^{\prime}(x) = f^{\prime}(x) + g^{\prime}(x)

Using this rule, we can find the derivative of the sum of $f$ and $g$ at $x=3$ :

(f+g)^{\prime}(3) = f^{\prime}(3) + g^{\prime}(3) = -5 + 6 = 1

Derivative of the Product of Two Functions

The derivative of the product of two functions is given by the product rule:

(fg)^{\prime}(x) = f^{\prime}(x)g(x) + f(x)g^{\prime}(x)

Using this rule, we can find the derivative of the product of $f$ and $g$ at $x=3$ :

(fg)^{\prime}(3) = f^{\prime}(3)g(3) + f(3)g^{\prime}(3) = (-5)(2) + (4)(6) = -10 + 24 = 14

Derivative of the Quotient of Two Functions

The derivative of the quotient of two functions is given by the quotient rule:

\left(\frac{f}{g}\right)^{\prime}(x) = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}

Using this rule, we can find the derivative of the quotient of $f$ and $g$ at $x=3$ :

\left(\frac{f}{g}\right)^{\prime}(3) = \frac{f^{\prime}(3)g(3) - f(3)g^{\prime}(3)}{(g(3))^2} = \frac{(-5)(2) - (4)(6)}{(2)^2} = \frac{-10 - 24}{4} = \frac{-34}{4} = -8.5

Conclusion

In this article, we have used the sum rule, product rule, and quotient rule to find the derivatives of composite functions. We have also used the given values to find the derivatives of the sum, product, and quotient of two functions at a specific point. These rules are essential in calculus and are used to find the derivatives of complex functions.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart

Glossary

Derivative: The rate of change of a function with respect to its input.
Sum Rule: The rule for finding the derivative of the sum of two functions.
Product Rule: The rule for finding the derivative of the product of two functions.
Quotient Rule: The rule for finding the derivative of the quotient of two functions.
Derivatives of Composite Functions: Q&A =====================================

In the previous article, we discussed the derivatives of composite functions using the sum rule, product rule, and quotient rule. In this article, we will answer some frequently asked questions about derivatives of composite functions.

Q: What is the derivative of the sum of two functions?

A: The derivative of the sum of two functions is given by the sum rule:

(f+g)^{\prime}(x) = f^{\prime}(x) + g^{\prime}(x)

Q: What is the derivative of the product of two functions?

A: The derivative of the product of two functions is given by the product rule:

(fg)^{\prime}(x) = f^{\prime}(x)g(x) + f(x)g^{\prime}(x)

Q: What is the derivative of the quotient of two functions?

A: The derivative of the quotient of two functions is given by the quotient rule:

\left(\frac{f}{g}\right)^{\prime}(x) = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2}

Q: How do I apply the sum rule, product rule, and quotient rule?

A: To apply the sum rule, product rule, and quotient rule, you need to follow these steps:

Identify the functions involved in the composite function.
Find the derivatives of each function using the power rule, product rule, and quotient rule.
Apply the sum rule, product rule, or quotient rule to find the derivative of the composite function.

Q: What are some common mistakes to avoid when finding derivatives of composite functions?

A: Some common mistakes to avoid when finding derivatives of composite functions include:

Forgetting to apply the power rule, product rule, or quotient rule.
Not using the correct notation for the derivative.
Not simplifying the expression after applying the sum rule, product rule, or quotient rule.

Q: How do I check my work when finding derivatives of composite functions?

A: To check your work when finding derivatives of composite functions, you can:

Use a calculator to check your answer.
Check your work by plugging in a specific value for the variable.
Check your work by using a different method to find the derivative.

Q: What are some real-world applications of derivatives of composite functions?

A: Derivatives of composite functions have many real-world applications, including:

Physics: Derivatives of composite functions are used to model the motion of objects and to calculate the force of gravity.
Engineering: Derivatives of composite functions are used to design and optimize systems, such as bridges and buildings.
Economics: Derivatives of composite functions are used to model the behavior of economic systems and to calculate the rate of change of economic variables.

Conclusion

In this article, we have answered some frequently asked questions about derivatives of composite functions. We have also discussed the sum rule, product rule, and quotient rule, and provided some tips for applying these rules. By following these tips and practicing with examples, you can become proficient in finding derivatives of composite functions.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart

Glossary

Derivative: The rate of change of a function with respect to its input.
Sum Rule: The rule for finding the derivative of the sum of two functions.
Product Rule: The rule for finding the derivative of the product of two functions.
Quotient Rule: The rule for finding the derivative of the quotient of two functions.