Sum Rule for finding Derivatives

The Sum Rule, also known as the Addition Rule, is a fundamental principle in differentiation used to find the derivative of a sum of two or more functions. This rule simplifies the process of differentiating such functions and is widely used in various applications of calculus, including optimization problems, physics, and engineering.

Steps to Apply the Sum Rule

1. Identify the functions: Determine the individual functions $f(x)$, $g(x)$, or any other functions in the given sum.
2. Find the derivatives of each function: Differentiate each function in the sum with respect to its variable using the appropriate differentiation rules.
3. Add the derivatives: Add the derivatives of the individual functions obtained in step 2.
4. Simplify the result: If necessary, simplify the final expression obtained from step 3.

Example: Applying the Sum Rule

Let’s consider the function $h(x)=x^3+\sin (x)$.

1. The functions in the sum are $f(x)=x^3$ and $g(x)=\sin (x)$.

2. The derivative of $x^3$ is $3x^2$ (using the Power Rule), and the derivative of $\sin (x)$ is $\cos (x)$.

3. Adding the derivatives:

   $$h^{\prime }(x)=\frac{\text{d}}{\text{d}x}(x^3+\sin (x))=\frac{\text{d}}{\text{d}x}{x}^3+\frac{\text{d}}{\text{d}x}\sin (x)=3x^2+\cos (x)$$

4. The result is already simplified, so we have:

   $$h^{\prime }(x)=3x^2+\cos (x)$$

Thus, the derivative of $h(x)=x^3+\sin (x)$ is $h^{\prime }(x)=3x^2+\cos (x)$.

Sum Rule Proof

To prove the Sum Rule for differentiation, we will use the definition of the derivative (i.e., the first principle of differentiation) and the limit properties. Let $h(x)=f(x)+g(x)$, where $f(x)$ and $g(x)$ are differentiable functions.

Proof:

Therefore, we have proved that $\frac{\text{d}}{\text{d}x}(f(x)+g(x))=\frac{\text{d}}{\text{d}x}f(x)+\frac{\text{d}}{\text{d}x}g(x)$.

Detailed Explanation of the Proof Steps

1. Start with the definition of $h(x)$: $h(x)$ is defined as the sum of $f(x)$ and $g(x)$:

   $$h(x)=f(x)+g(x)$$

2. Definition of the derivative: The derivative of $h(x)$ with respect to $x$ is given by:

   $$h^{\prime }(x)=\frac{\text{d}}{\text{d}x}(f(x)+g(x))$$

3. Apply the limit definition of the derivative: Using the limit definition, the derivative can be written as:

   $$h^{\prime }(x)={lim}_{h\to 0}\frac{(f(x+h)+g(x+h))-(f(x)+g(x))}{h}$$

4. Distribute the terms in the numerator: Rewrite the expression inside the limit by distributing the terms:

   $$h^{\prime }(x)={lim}_{h\to 0}\frac{(f(x+h)+g(x+h))-f(x)-g(x)}{h}$$

5. Separate the fractions: Split the fraction into two separate fractions:

   $$h^{\prime }(x)={lim}_{h\to 0}[\frac{f(x+h)-f(x)}{h}+\frac{g(x+h)-g(x)}{h}]$$

6. Use the sum of limits property: Apply the property that the limit of a sum is the sum of the limits:

   $$h^{\prime }(x)={lim}_{h\to 0}\frac{f(x+h)-f(x)}{h}+{lim}_{h\to 0}\frac{g(x+h)-g(x)}{h}$$

7. Recognize the individual derivatives: Each term inside the limits represents the derivative of the respective functions:

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

Thus, we have proved that: