Leibniz Rule Calculator

Formula

To calculate the derivative of the product of two functions using the Leibniz Rule:

\[ (f \cdot g)' = f' \cdot g + f \cdot g' \]

Where:

\((f \cdot g)'\) is the derivative of the product of function \(f\) and function \(g\)
\(f'\) is the derivative of function \(f\)
\(g\) is the original function \(g\)
\(f\) is the original function \(f\)
\(g'\) is the derivative of function \(g\)

What is the Leibniz Rule?

The Leibniz Rule, also known as the product rule, is a fundamental theorem in calculus that provides a formula to differentiate the product of two functions. Named after Gottfried Wilhelm Leibniz, this rule states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function. It is a key tool in differential calculus and is used to simplify the process of finding derivatives.

Example Calculation 1

Let's assume the following values:

Function \(f(x)\) = \(x^2\)
Function \(g(x)\) = \(\sin(x)\)
Derivative of \(f(x)\) (\(f'(x)\)) = \(2x\)
Derivative of \(g(x)\) (\(g'(x)\)) = \(\cos(x)\)

Using the formula:

\[ (f \cdot g)' = (2x \cdot \sin(x)) + (x^2 \cdot \cos(x)) = 2x \sin(x) + x^2 \cos(x) \]

The derivative of the product is \(2x \sin(x) + x^2 \cos(x)\).

Example Calculation 2

Let's assume the following values:

Function \(f(x)\) = \(e^x\)
Function \(g(x)\) = \(\ln(x)\)
Derivative of \(f(x)\) (\(f'(x)\)) = \(e^x\)
Derivative of \(g(x)\) (\(g'(x)\)) = \(\frac{1}{x}\)

Using the formula:

\[ (f \cdot g)' = (e^x \cdot \ln(x)) + (e^x \cdot \frac{1}{x}) = e^x \ln(x) + e^x \cdot \frac{1}{x} \]

The derivative of the product is \(e^x \ln(x) + e^x \cdot \frac{1}{x}\).