step by step + graph

Turning Points, Inflexion

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

First and Second Derivatives: Get the rate of change and concavity for any function you enter.

Turning Points: Automatically calculate and classify turning points (local minima, maxima, or saddle points) using Newton-Raphson methods.

Points of Inflection: Find inflection points where the function changes concavity, by solving \( f''(x) = 0 \).

Generate Interactive Graphs: The graph shows the original function, its derivatives, and highlights key points (turning points and points of inflection) for a better understanding of the function’s behavior.

Supported Expressions

Enter the following types of expressions into the input box

Polynomial Functions:

Example: x^2
Example: x^3 - 6*x^2 + 9*x + 2

Trigonometric Functions:

Example: sin(x)
Example: cos(x)
Example: tan(x)

Exponential Functions:

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

Logarithmic Functions:

Example: log(x)

Rational Functions:

Example: 1/x

Square Roots and Radicals:

Example: sqrt(x)

Combination of Functions:

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

CalculusPop differentiation solver helps you analyze any function through differentiation, providing insights into its behavior with:

First Derivative: Understand the rate of change of your function.

Second Derivative: Explore concavity and acceleration.

Turning Points: Identify local minima, maxima, and saddle points.

Points of Inflection: Locate where the function changes concavity.

How to Use It

Enter Your Function:

In the input box, type your function using standard mathematical notation. Check the examples below to see the correct formats for different types of functions.

Click "Solve":

The solver will compute and display the first and second derivatives. It will also find turning points and points of inflection, along with an interactive graph of the function and its derivatives.

Review the Steps: A detailed step-by-step explanation of the differentiation process is provided so you can understand each phase of the calculation.

Interactive Graph

The graph shows the original function, its derivatives, and highlights key points (turning points and points of inflection) for a better understanding of the function’s behavior.

First Derivative

The first derivative, denoted as \( f'(x) \), represents the rate of change of the function. Think of it as how quickly the output of a function changes as the input varies. For example, in physics, the first derivative of position with respect to time gives the velocity. Mathematically, it helps identify slopes at any given point on a curve. If \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), it’s decreasing.

Second Derivative

The second derivative, denoted \( f”(x) \), goes one step further by measuring the rate of change of the rate of change. In simpler terms, it tells us about the concavity of the graph. If \( f”(x) > 0 \), the function is concave up (shaped like a cup). If \( f”(x) < 0 \), it’s concave down (shaped like a cap). In physics, this concept is analogous to acceleration—how fast velocity itself is changing.

Turning Points

 Turning points are where a function switches direction from increasing to decreasing or vice versa. These can be either local maxima (where the function reaches a peak) or local minima (where it dips to a valley). Mathematically, turning points occur where the first derivative equals zero, i.e., \( f'(x) = 0 \). The second derivative helps classify them: if \( f”(x) > 0 \), it’s a minimum; if \( f”(x) < 0 \), it’s a maximum.

Points of Inflection

A point of inflection is where a function’s concavity changes. This is crucial for understanding the overall shape of a curve. For example, a function might switch from concave up (cup-shaped) to concave down (cap-shaped). These points occur where the second derivative equals zero, \( f”(x) = 0 \), and the concavity changes sign. Unlike turning points, at points of inflection, the function does not necessarily reach a peak or a valley—it’s simply a change in how the curve bends.

 

Polynomial Functions

Polynomial functions are algebraic expressions involving sums of powers of \( x \). They’re easy to differentiate because the power rule applies: \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \). For example, the derivative of \( x^3 \) is \( 3x^2 \). Polynomial functions are common in everyday applications, from physics to economics, because they represent smooth, continuous behaviors.

Trigonometric Functions

Trigonometric functions like \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \) are crucial in modeling periodic phenomena, such as sound waves or seasons. The derivatives of these functions are cyclical too. For example, \( \frac{d}{dx}(\sin(x)) = \cos(x) \) and \( \frac{d}{dx}(\cos(x)) = -\sin(x) \). These relationships are foundational in trigonometry and calculus, especially in solving wave equations or analyzing oscillations.

Exponential Functions

Exponential functions, like \( e^x \), model growth and decay processes, such as population growth or radioactive decay. What makes them unique is that the rate of change of \( e^x \) is proportional to the function itself: \( \frac{d}{dx}(e^x) = e^x \). This property is key in fields like biology and finance, where exponential models describe how systems grow over time.

 Logarithmic Functions

The natural logarithm, \( \log(x) \), is the inverse of the exponential function. It helps us solve equations where the unknown appears as an exponent. The derivative of \( \log(x) \) is simple yet powerful: \( \frac{d}{dx}(\log(x)) = \frac{1}{x} \). This formula is often used when working with rates of growth or when simplifying complex expressions involving multiplication or division.

Rational Functions

Rational functions are ratios of polynomials, like \( \frac{1}{x} \). They often appear in real-world problems involving rates, such as speed or density. The derivative of a rational function can be found using the quotient rule, which helps break down the relationship between the numerator and the denominator. For example, \( \frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2} \).

Square Roots and Radicals

Square roots, like \( \sqrt{x} \), are a special type of power function where the exponent is a fraction. The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \), using the general power rule \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \). These functions come up in various areas, from geometry to physics, when calculating distances, areas, or solving quadratic equations.

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