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Product Rule Derivative Calculator

Write down the equation, select the variable, and choose the order of derivation. The tool will instantly determine the derivative, providing detailed step-by-step calculations.

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Product Rule Derivative Calculator

The online Product Rule Derivative Calculator helps you find the derivative of functions that are products of differentiable functions. Using the product rule of differentiation, this tool quickly simplifies calculations and provides step-by-step results.

What Is the Product Rule?

In calculus, the product rule is applied when two or more functions are multiplied together. For two differentiable functions f(x) and g(x), the product rule states:

Derivative of f(x) × g(x) = f(x) × g'(x) + g(x) × f'(x)

Formula:

For differentiable functions f(x) and g(x):

$$ \frac{d}{dx}[f(x) g(x)] = f(x) \frac{d}{dx}g(x) + g(x) \frac{d}{dx}f(x) $$

Instead of computing manually, you can use the online product rule derivative calculator for instant results.

How to Apply the Product Rule

Example 1:

Differentiate the function:

$$ h(x) = (6x^2 - x)(1 - 30x) $$

Solution:

Let:

$$ f(x) = 6x^2 - x, \quad g(x) = 1 - 30x $$

Derivatives:

$$ f'(x) = 12x - 1, \quad g'(x) = -30 $$

Applying the product rule:

$$ h'(x) = f(x) g'(x) + g(x) f'(x) $$

$$ = (6x^2 - x)(-30) + (1 - 30x)(12x - 1) $$

$$ = -540x^2 + 72x - 1 $$

Example 2:

Differentiate:

$$ h(z) = (z^2)^{1/3} (2z - z^2) $$

Solution:

Let:

$$ f(z) = (z^2)^{1/3}, \quad g(z) = 2z - z^2 $$

Derivatives:

$$ f'(z) = \frac{2 (z^2)^{1/3}}{3z}, \quad g'(z) = 2 - 2z $$

Using the product rule:

$$ h'(z) = f(z) g'(z) + g(z) f'(z) $$

$$ = (z^2)^{1/3} (2 - 2z) + (2z - z^2) \frac{2 (z^2)^{1/3}}{3z} $$

$$ = \frac{2 (5 - 4z) (z^2)^{1/3}}{3} $$

How the Calculator Works

Input:

Enter the function using supported operations (e.g., log, ln, sqrt, sin, cos, tan)
Select the variable to differentiate (e.g., x, y, z)
Choose the differentiation order (up to 5)
Click “Calculate”

Output:

Derivative of the function using the product rule
Step-by-step simplification
Fully simplified result

FAQs

What is the product rule of exponents?

When multiplying exponential expressions with the same base, add the exponents:

$$ a^m \cdot a^n = a^{m+n} $$

Example: $$ a^5 \cdot a^8 = a^{13} $$

Can the product rule be applied to more than 2 terms?

Yes. Differentiate each function separately and sum the results accordingly.

What is the natural logarithm of zero?

The natural logarithm ln(x) is defined for x > 0 only. Hence, ln(0) is undefined (approaches -∞).

What is the derivative of log(e)?

Since log(e) = 1, its derivative is:

$$ \frac{d}{dx}[1] = 0 $$

Conclusion

The product rule is essential for differentiating products of functions in calculus and engineering. Using an online product rule derivative calculator saves time, avoids errors, and provides step-by-step solutions for both students and professionals.

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