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Leibniz Rule Calculator Online

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The Leibniz Rule Calculator computes the nth derivative of the product of two functions using the Leibniz rule. This rule is essential in calculus for handling higher-order derivatives of products, making it a valuable resource for solving complex mathematical problems.

Formula of Leibniz Rule Calculator

The Leibniz rule is used to find the nth derivative of a product of two functions. The formula is:

Leibniz Rule

where:

  • n is the order of the derivative
  • C(n, k) is the binomial coefficient, given by C(n, k) = n! / (k! * (n - k)!)
  • (d^k/dx^k) v(x) is the k-th derivative of v(x)
  • (d^(n - k)/dx^(n - k)) u(x) is the (n - k)-th derivative of u(x)
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This formula helps in finding higher-order derivatives of products of functions.

Common Terms and Conversions Table

Below is a table of commonly searched terms related to the Leibniz rule and their corresponding explanations:

TermExplanation
DerivativeThe rate at which a function is changing at any given point
Higher-order DerivativeDerivatives of derivatives, indicating rates of change of rates of change
Binomial Coefficient (C(n,k))A coefficient giving the number of ways to choose k elements from n elements
Product of FunctionsA function that is the result of multiplying two other functions
nth DerivativeThe derivative taken n times

Example of Leibniz Rule Calculator

Consider the functions u(x) = x^2 and v(x) = e^x. We want to find the 3rd derivative of their product using the Leibniz rule.

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Using the formula:

d^3/dx^3 [x^2 e^x] = sum from k=0 to 3 of C(3, k) * (d^(3-k)/dx^(3-k)) x^2 * (d^k/dx^k) e^x

Calculating each term:

For k = 0: C(3, 0) * (d^3/dx^3) x^2 * e^x = 1 * 0 * e^x = 0

For-k = 1: C(3, 1) * (d^2/dx^2) x^2 * (d/dx) e^x = 3 * 2 * e^x = 6e^x

For k = 2: C(3, 2) * (d/dx) x^2 * (d^2/dx^2) e^x = 3 * 2x * e^x = 6xe^x

For-k = 3: C(3, 3) * x^2 * (d^3/dx^3) e^x = 1 * x^2 * e^x = x^2 e^x

Summing all terms: d^3/dx^3 [x^2 e^x] = 0 + 6e^x + 6xe^x + x^2 e^x

Most Common FAQs

Why is the Leibniz rule important?

The Leibniz rule is crucial in calculus as it simplifies the process of finding higher-order derivatives of product functions. This is especially useful in fields requiring precise mathematical modeling and analysis.

Can the Leibniz Rule Calculator handle any functions?

Yes, the calculator is designed to handle a wide range of functions as long as they are properly defined and differentiable. This includes polynomial, exponential, trigonometric, and logarithmic functions, among others.

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