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Derivative Calculator

z.B. x^2 + sin(x)

💡 Klicken Sie auf die Tasten, um mathematische Symbole einzufügen

Dies ist ein KI-gestützter Ableitungsrechner. Die Berechnung komplexer Ausdrücke kann einige Sekunden dauern.

Why Use Our Derivative Calculator?

Discover the benefits of our powerful tool

Lightning-Fast Calculation

Instant results for complex derivatives

Step-by-Step Solutions

Detailed explanation of the solution path

Versatile Function Support

Supports a wide range of functions

100% Free and Secure

No hidden costs, your data is safe

How to Use the Derivative Calculator

1

Enter Function

Enter the mathematical function you want to differentiate in the input field. The derivative calculator supports various notations. For example, you can write x^2 for x² or sqrt(x) for the square root. Make sure to mark multiplications with *, so 2*x instead of 2x.

2

Select Variable

Choose the variable with respect to which you want to differentiate. In most cases, this is x. However, the derivative calculator can also differentiate with respect to other variables like y, t, or any other letter if your function contains multiple variables.

3

Start Calculation

Click the "Calculate Derivative" button and the derivative calculator will instantly determine the derivative of your function. The calculation is done in real-time and you will immediately see the result as well as all intermediate steps of the calculation.

4

Review Result

Review the result and individual calculation steps. The derivative calculator shows you exactly which derivative rules were applied, so you can follow the solution process. You can copy the result directly or use it for further calculations.

Derivative Examples

Polynomial Function

Function: f(x) = x³ + 2x² - 5x + 3

Derivative: f'(x) = 3x² + 4x - 5

Trigonometric Function

Function: f(x) = sin(x) + cos(x)

Derivative: f'(x) = cos(x) - sin(x)

Exponential Function

Function: f(x) = e^x + 2^x

Derivative: f'(x) = e^x + 2^x · ln(2)

Logarithmic Function

Function: f(x) = ln(x) + log(x)

Derivative: f'(x) = 1/x + 1/(x·ln(10))

Chain Rule

Function: f(x) = sin(x²)

Derivative: f'(x) = 2x · cos(x²)

Product Rule

Function: f(x) = x · e^x

Derivative: f'(x) = e^x + x · e^x

What is a Derivative?

The derivative of a function describes the rate of change or slope of the function at every point. It is one of the most fundamental concepts in calculus and has numerous applications in science, engineering, and economics.

Mathematically, the derivative is defined as the limit of the difference quotient:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

In practice, however, derivative rules are used to calculate derivatives quickly and efficiently.

Important Derivative Rules

Power Rule

(xn)′=n⋅xn−1(x^n)' = n \cdot x^{n-1}

Sum Rule

(f+g)′=f′+g′(f + g)' = f' + g'

Product Rule

(f⋅g)′=f′⋅g+f⋅g′(f \cdot g)' = f' \cdot g + f \cdot g'

Quotient Rule

(f/g)′=f′g−fg′g2(f/g)' = \frac{f'g - fg'}{g^2}

Chain Rule

(f(g(x)))′=f′(g(x))⋅g′(x)(f(g(x)))' = f'(g(x)) \cdot g'(x)

Exponential

(ex)′=ex(e^x)' = e^x

Logarithmus

(ln⁡x)′=1x(\ln x)' = \frac{1}{x}

Sinus

(sin⁡x)′=cos⁡x(\sin x)' = \cos x

Cosine

(cos⁡x)′=−sin⁡x(\cos x)' = -\sin x

Frequently Asked Questions About the Derivative Calculator

What is a derivative calculator and how does it work?

A derivative calculator is a mathematical tool that automatically computes the derivative of a given function. Our tool uses advanced algorithms to apply derivative rules such as power rule, product rule, quotient rule, and chain rule. It analyzes the input function, recognizes mathematical structures, and delivers the correct result.

What types of functions can the derivative calculator process?

The derivative calculator can differentiate a wide variety of mathematical functions: polynomial functions (like x² + 3x + 2), trigonometric functions (sin, cos, tan), exponential functions (e^x, a^x), logarithmic functions (ln, log), root functions, and complex composite functions. The derivative calculator masters chain rule, product rule, and quotient rule.

Is the derivative calculator free?

Yes, our derivative calculator is completely free to use without any limitations. You can use the derivative calculator as often as you want, without registration or fees. The derivative calculator is available to you around the clock.

Does the derivative calculator show the calculation steps?

Yes, the derivative calculator shows not only the final result but also all intermediate steps of the calculation. This allows you to understand exactly which derivative rules were applied. This feature makes the derivative calculator a valuable learning tool.

Can I calculate higher-order derivatives with the derivative calculator?

Yes, the derivative calculator can compute not only the first derivative but also higher-order derivatives such as the second, third, or nth derivative. This is particularly useful for applications in physics and engineering.

How accurate is the derivative calculator?

The derivative calculator works with symbolic mathematics and therefore delivers exact analytical results, not just numerical approximations. You can rely on the correctness of the results that the derivative calculator provides.

Is registration required?

No, you can use the tool immediately without registration. No sign-up is required and you don't need to provide any personal information. Simply open the website, enter your function, and instantly get the result.

Does it work on smartphones?

Yes, our derivative calculator is fully responsive and works on all devices - whether desktop computer, tablet, or smartphone. The design automatically adapts to your screen size, so you can use it conveniently from anywhere.

Which derivative rules are applied?

The system masters all basic and advanced derivative rules: power rule for polynomials, product rule for products of functions, quotient rule for fractions, chain rule for composite functions, as well as special rules for trigonometric, exponential, and logarithmic functions. The appropriate rule is automatically selected.

Is this tool suitable for students?

Yes, it is an ideal tool for students. It not only helps with homework but also serves as a learning aid by displaying all calculation steps. This way, derivative rules can be better understood and applied independently. Many teachers recommend it as a verification tool.

Essential Knowledge About Derivatives

Understand the fundamentals and practical applications of differential calculus

What is a Derivative?

The derivative describes the rate of change of a function. Geometrically, it represents the slope of the tangent line at a point. For f(x) = x², the derivative is f'(x) = 2x.

Applications in Physics

In physics, the first derivative of position represents velocity, the second derivative represents acceleration. Essential for motion analysis.

Important Derivative Rules

Power rule: (x^n)' = n·x^(n-1). Product rule: (u·v)' = u'·v + u·v'. Chain rule: (f∘g)' = (f'∘g)·g'. These rules are applied automatically.

Economic Applications

Calculation of marginal cost, marginal revenue, and optimization. Finds the point of maximum profit or minimum cost through zeros of the derivative.

💡Tips for Optimal Use

  • •Use x^2 for x² and * for multiplication
  • •Use example functions for quick testing
  • •Study the calculation steps for better understanding
  • •Visualize results with the integrated function graph