### Differential Calculator Uses

**1. Constant Rule**

The derivative of a constant is always zero.

Example: \( \frac{d}{dx}(5) = 0 \).

**2. Power Rule**

For any power of \( x \), \( \frac{d}{dx}(x^n) = nx^{n-1} \).

Example: \( \frac{d}{dx}(x^3) = 3x^2 \).

**3. Sum and Difference Rule**

The derivative of a sum or difference of functions is the sum or difference of their derivatives.

Example: \( \frac{d}{dx}(x^2 + 3x - 1) = 2x + 3 \).

**4. Product Rule**

For two functions \( u(x) \) and \( v(x) \), the derivative is given by:

\[

\frac{d}{dx}[u \cdot v] = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}

\]

Example: \( \frac{d}{dx}(x \cdot e^x) = x \cdot e^x + e^x \).

**5. Quotient Rule**

For a quotient of two functions \( u(x) \) and \( v(x) \), the derivative is:

\[

\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}

\]

Example: \( \frac{d}{dx}\left( \frac{2x}{x+1} \right) = \frac{(x+1)(2) - 2x(1)}{(x+1)^2} \).

**6. Chain Rule (Composite Functions)**

For a composite function \( f(g(x)) \), the chain rule states:

\[

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

\]

Example: \( \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot 2 \).

**7. Trigonometric Functions**

Sine: \( \frac{d}{dx}(\sin(x)) = \cos(x) \)

Cosine: \( \frac{d}{dx}(\cos(x)) = -\sin(x) \)

Tangent: \( \frac{d}{dx}(\tan(x)) = \sec^2(x) \)

**8. Exponential Functions**

For any exponential function \( e^x \), the derivative is:

\[

\frac{d}{dx}(e^x) = e^x

\]

Example: \( \frac{d}{dx}(e^{2x}) = 2e^{2x} \).

**9. Logarithmic Functions**

For the natural logarithm, the derivative is:

\[

\frac{d}{dx}(\log(x)) = \frac{1}{x}

\]

Example: \( \frac{d}{dx}(\log(x^2)) = \frac{2}{x} \).

**10. Higher-Order Derivatives**

The solver can also calculate higher-order derivatives (e.g., the second derivative, third derivative, etc.).

Example: The second derivative of \( x^3 \) is \( \frac{d^2}{dx^2}(x^3) = 6x \).

**11. Exponents and Roots**

For functions involving square roots or other fractional exponents:

\[

\frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2}

\]

Example: \( \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \).

**12. Inverse Trigonometric Functions**

Arcsin: \( \frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}} \)

Arccos: \( \frac{d}{dx}(\cos^{-1}(x)) = \frac{-1}{\sqrt{1 - x^2}} \)

Arctan: \( \frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1 + x^2} \)

1. Constant Expressions: Easily computes the derivative of constant terms, which always yield zero (e.g., \( 7 \)).

2. Single Variable Differentiation: Handles basic differentiation of single variables like \( x \), where \( \frac{d}{dx}(x) = 1 \).

3. Linear Expressions: Solves linear equations like \( ax \), providing instant results (e.g., \( 5x \) yields 5).

4. Power Rule: Applies the power rule to expressions like \( x^n \) (e.g., \( x^4 \) becomes \( 4x^3 \)).

5. Sum or Difference of Functions: Differentiates terms added or subtracted (e.g., \( x^2 + 3x - 1 \)).

6. Product Rule: Solves expressions where two functions are multiplied (e.g., \( x \cdot e^x \)).

7. Quotient Rule: Differentiates complex quotients (e.g., \( \frac{3x}{x^2 + 1} \)).

8. Chain Rule for Composite Functions: Handles the differentiation of nested functions (e.g., \( \sin(2x) \) or \( \log(4x^2) \)).

9. Trigonometric Functions: Works with functions like \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), and more.

10. Exponential and Logarithmic Functions: Differentiates expressions such as \( e^x \) and \( \log(x) \).

1. 5

2. x

3. 4x

4. x^3

5. x^2 + 3x - 1

6. x^2 * e^x

7. (2x) / (x^2 + 1)

8. sin(2x)

9. log(3x^2)

10. cos(x)

11. e^x

12. tan(x^3)

13. x^4 + 5x^3 - 2x + 8

14. exp(2x)

15. 3x / (x^2 + 4)

16. x^5 * log(x)

17. (x^2 + 2x + 1)^3

18. cos(4x) * e^x

19. log(x) / x^2

20. sin^2(x)

21. e^(x^2)

22. 1 / (x^2 + 1)

23. 3x^2 - 4x + 6

24. tan(3x) * log(x)

25. (x + 1) / (x^2 - x)

26. x^2 * sin(x)

27. log(x^2 + 3x)

28. exp(x) * cos(x)

29. (x^2 - 4) / (x + 2)

30. sqrt(x) * log(x)

#### Worked Example 1

Expression: x^3 + 4x^2 - 5x + 2

**Input in Raw Form**

x^3 + 4x^2 - 5x + 2

#### Step-by-Step Output

1. Step 1: Apply the power rule to each term individually.

- \( \frac{d}{dx}(x^3) = 3x^2 \)

- \( \frac{d}{dx}(4x^2) = 8x \)

- \( \frac{d}{dx}(-5x) = -5 \)

- \( \frac{d}{dx}(2) = 0 \)

2. Step 2: Combine the results.

- The derivative is:

\[

\frac{d}{dx}(x^3 + 4x^2 - 5x + 2) = 3x^2 + 8x - 5

\]

#### Worked Example 2

Expression:** sin(2x)**

Input in Raw Form:

sin(2x)

#### Step-by-Step Output

1. Step 1: Recognize the composite function and apply the chain rule.

- Let \( u = 2x \), so \( \frac{d}{dx}(sin(2x)) = cos(2x) \cdot \frac{d}{dx}(2x) \).

2. Step 2: Differentiate the inner function.

- \( \frac{d}{dx}(2x) = 2 \).

3. Step 3: Multiply the results.

- The derivative is:

\[

\frac{d}{dx}(sin(2x)) = 2 \cdot cos(2x)

\]

### How to Use Differentiation Solver

**1. Enter Your Expression**

Start by typing your mathematical expression into the input box. You can solve anything from simple variables to more intricate functions like cos(x^2) or exp(x).

As you type, the expression will be rendered using KaTeX for easy visualization, allowing you to verify the structure before proceeding.

**2. Differentiate the Expression**

After inputting your equation, hit the Differentiate button. Instantly, CalculusPop will solve the differential equation and provide a step-by-step explanation, making it perfect for learners or professionals who need to follow the solution process.

Whether you’re solving for academic purposes or working on complex real-world problems, this differential eq. solver walks you through each calculation.

**3. Detailed Steps**

CalculusPop doesn’t just provide answers; it walks you through every step of the differentiation process. This feature is particularly helpful if you’re trying to understand how to solve differential equations methodically. You’ll see explanations for key concepts like the product rule, chain rule, and power rule, all clearly displayed for easy understanding.

**4. Graphing for Visualization**

Once the differentiation is complete, a graph is generated to help you visualize both the original function and its derivative. This interactive graphing feature offers valuable insights into the behavior of your function and its rate of change, allowing for deeper exploration of your results.

#### Example

Let’s say you input the equation \( 2x^3 + 3x – 4 \):

The equation will be rendered clearly as \( 2x^3 + 3x – 4 \).

The differentiation process will then be displayed step-by-step, showing how the derivative is calculated as \( 6x^2 + 3 \).

A graph will be generated, plotting both \( 2x^3 + 3x – 4 \) and its derivative \( 6x^2 + 3 \), giving you a visual representation of the function’s behavior over various values of \( x \).

By breaking down the process this way, CalculusPop isn’t just a tool; it’s an interactive learning experience, helping you grasp the nuances of differential calculus. Whether you’re studying, working on research, or simply refining your skills, this differentiation solver with steps ensures that every equation is approached logically, with clarity at every turn.

CalculusPop differentiation equation solver stands out from generic calculators and tools by emphasizing not just solving problems, but also teaching users how to solve differential equations from first principles. With this tool, you can confidently tackle even the most complex expressions, all while gaining deeper insights into the world of calculus.