\frac{dy}{dx} = a \cdot n \cdot x^{n-1}

The Continuous Differentiation Solver is designed to compute:

**First Derivative**

Understand the rate of change and identify whether the function is increasing or decreasing.

**Second Derivative**

Explore the concavity of the function and locate points of inflection.

**Higher-Order Derivatives**

Continue finding successive derivatives (third, fourth, etc.) to analyze the behavior of the function in more detail.

**Turning Points**

Automatically identify and classify turning points (local minima and maxima).

**Points of Inflection**

Discover where the function changes concavity.

1 Accurate Results

Solve derivatives instantly with precision.

**2 Step-by-Step Explanations**

Follow along with clear, detailed steps that show how each derivative is computed.

**3 Interactive Graphing**

Visualize the function and its derivatives to gain deeper insights into its behavior.

**4 Wide Range of Expressions**

Solve polynomial, trigonometric, logarithmic, exponential, and rational functions, along with combinations of these functions.

The calculator supports a wide variety of functions. You can input expressions in the following raw formats:

**Polynomial Functions**

x^2, x^3 - 3*x^2 + 2*x

**Trigonometric Functions**

sin(x), cos(x), tan(x)

**Exponential Functions**

e^x, a^x (replace a with any constant, e.g., 2^x)

**Logarithmic Functions**

log(x)

**Rational Functions**

1/x

**Square Roots and Radicals**

sqrt(x)

**Combination of Functions**

x^2 * sin(x) + e^x

**Continuous Differentiation Solver**, your go-to tool for solving higher-order derivatives with ease. This powerful calculator provides step-by-step solutions, helping you understand every phase of differentiation. Whether you're a student learning calculus or a professional needing quick results, this tool simplifies the process and gives you the ability to solve complex expressions instantly.

CalculusPop differentiation solver handles both simple and complex functions, computing first, second, and higher-order derivatives. The tool applies the most relevant differentiation rules—such as the power rule, product rule, quotient rule, and chain rule—automatically, allowing you to focus on understanding the results.

Not only does it give the exact derivative of any function, but it also provides a clear step-by-step explanation. The interactive graph allows you to visualize the original function and its derivatives, offering better insights into turning points, points of inflection, and concavity.

**Enter Your Expression**:

Type your function into the input box. The solver accepts a wide range of mathematical expressions.**Set the Iterations**:

Choose how many times you want to differentiate the function by entering the number of iterations. This will determine whether the tool calculates the first, second, third, or higher-order derivatives.**Solve and Explore**:

Click the "Solve" button to calculate the derivative. The tool will display the solution step-by-step, helping you understand the process of each successive differentiation.**Graph Visualization**:

The interactive graph displays the original function and its derivatives. You can zoom, pan, and interact with the graph to explore the behavior of the function visually.

Let’s take the function \( f(x) = x^3 – 3x^2 + 2x \) and find its first, second, and third derivatives.

**Step 1: First Derivative \( f'(x) \)**

To differentiate \( f(x) = x^3 – 3x^2 + 2x \), we apply the power rule to each term:

– The derivative of \( x^3 \) is \( 3x^2 \).

– The derivative of \( -3x^2 \) is \( -6x \).

– The derivative of \( 2x \) is \( 2 \).

Thus, the first derivative is:

\[

f'(x) = 3x^2 – 6x + 2

\]

This derivative tells us the rate of change of the function and where it increases or decreases.

**Step 2: Second Derivative \( f”(x) \)**

Now, we differentiate \( f'(x) = 3x^2 – 6x + 2 \) to find the second derivative:

– The derivative of \( 3x^2 \) is \( 6x \).

– The derivative of \( -6x \) is \( -6 \).

– The derivative of the constant \( 2 \) is \( 0 \).

Thus, the second derivative is:

\[

f”(x) = 6x – 6

\]

This second derivative tells us about the concavity of the function and helps identify points of inflection.

**Step 3: Third Derivative \( f”'(x) \)**

Finally, we differentiate \( f”(x) = 6x – 6 \) to find the third derivative:

– The derivative of \( 6x \) is \( 6 \).

– The derivative of \( -6 \) is \( 0 \).

Thus, the third derivative is:

\[

f”'(x) = 6

\]

The third derivative, in this case, is constant, meaning the function has no further changes in acceleration. In real-world applications, this could represent a uniform change in motion (e.g., a constant force).