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).
Successive differentiation refers to the process of differentiating a function multiple times. After finding the first derivative, you can continue finding higher-order derivatives (second, third, etc.). Each successive derivative provides additional insights into the behavior of the function, such as concavity or jerk (in physics). This process is essential for understanding changes in acceleration or analyzing more complex behaviors in a function.
The first derivative, \( f'(x) \), is the rate of change of the function. It shows how the function’s output changes with respect to its input. This derivative is used to find slopes and tells you whether a function is increasing or decreasing. It is the foundation for further successive differentiation.
The second derivative, \( f”(x) \), gives the rate of change of the first derivative. It helps determine the concavity of the function, indicating whether the function is curving upwards (concave up) or downwards (concave down). It is especially useful in locating points of inflection, where the concavity changes.
After the second derivative, you can continue differentiating to find the third derivative, \( f”'(x) \), and beyond. In applied contexts like physics, higher-order derivatives are used to describe physical phenomena. For example, the third derivative of a position function represents jerk, or the rate at which acceleration changes. While these derivatives may not always have simple physical interpretations, they are valuable in analyzing more complex function behaviors.
The power rule is a basic but powerful tool for differentiating polynomials. For a function of the form \( x^n \), the derivative is \( n \cdot x^{n-1} \). Successive differentiation of powers of \( x \) reduces the exponent until it reaches zero, at which point the derivative becomes zero. This explains why higher-order derivatives of simple polynomials eventually disappear.
When differentiating products of two functions, the product rule is applied. For two functions \( u(x) \) and \( v(x) \), the product rule is \( (uv)’ = u’v + uv’ \). Successive differentiation may require the repeated use of the product rule to differentiate combinations of functions.
The quotient rule is used when differentiating the ratio of two functions, \( \frac{u(x)}{v(x)} \). The formula is \( \left(\frac{u}{v}\right)’ = \frac{v u’ – u v’}{v^2} \). Successive differentiation of quotients can become more complicated, but the quotient rule helps systematically differentiate these expressions.
The chain rule is crucial when differentiating composite functions. If \( y = f(g(x)) \), then the derivative is \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \). This rule becomes increasingly important in successive differentiation when functions are nested, and the derivatives of both the inner and outer functions must be taken.