Sum and difference rule in differentiation

Differentiation is the mathematical process of finding the rate at which a function changes with respect to its independent variable. The sum rule in differentiation states that the derivative of the sum of two functions is equal to the sum of their derivatives. Similarly, the difference rule states that the derivative of the difference between two functions is equal to the difference of their derivatives.

You can see here for more details of other differentiation rules.

Sum and difference rule mathematically

Mathematically, the sum rule or difference rule states that the derivative of the sum or differences of two functions is equal to the sum of their individual derivatives:

Let's say you have two functions, $f(x)$ and $g(x)$, and you want to find the derivative of their sum, $h(x) = f(x) \pm g(x)$.

This rule can be extended to more than two functions. For example, if you have three functions $h(x) = f(x) \pm g(x) \pm k(x)$, then the derivative of $h(x)$ with respect to x would be:

Examples

We will see a few examples to see how the sum or difference rule in differentiation works.

Example 1

Question: Differentiate the function $h(x) = 10x^{3} + 2x$.

Answer: According to the sum rule of differentiation, we will derivate each component of this function separately.$h(x) = f(x) \pm g(x),f(x)=10x^{3} \text{ and } g(x) = 2x$

Example 2

Question: Differentiate the function $h(x) = sin(x) +10x^{2} - 40x$.

Answer: According to the sum rule and difference rule of differentiation, we will derivate each component of this function separately. In this example, we can identify a total of three separate functions.

$h(x) = f(x) \pm g(x) \pm k(x),f(x) = sin(x) ,g(x) = 10x^{2} ,\text{and } k(x) = -40x$

Quiz

Now that you know the sum and difference rules of differentiation, you can challenge yourself with a quiz. Remember to differentiate each function separately.

Conclusion

Both the sum rule and difference rule of differentiation allow us to find the derivative of the sum or difference of any number of functions, not limited to just two. It simplifies the process of calculating derivatives for more generalized mathematical expressions.