When should I use the Product Rule vs Chain Rule?

Use the **Product Rule** when two variable expressions are multiplied side-by-side, like $x \cdot \sin(x)$. Use the **Chain Rule** when one function is inside another, like $\sin(x^2)$. Sometimes you need both, like in $x^2 \sin(x^3)$.

Why does the result look different from my textbook?

Mathematical expressions can often be written in multiple equivalent forms (e.g., $2(x+1)$ vs $2x+2$). Our calculator produces a mathematically correct derivative, but its simplification level or term ordering might differ from your textbook's preference. You can verify equivalence by plugging in a specific value (e.g., $x=1$) into both expressions.

Can I just multiply the derivatives?

No! This is a very common mistake. The derivative of a product is **not** the product of the derivatives. You must use the formula $f'g + fg'$.

What if there are three functions, like $f(x)g(x)h(x)$?

You can apply the Product Rule twice, or use the extended formula: $(fgh)' = f'gh + fg'h + fgh'$. Basically, you differentiate one function at a time while keeping the others the same, then add them all up.

How do I handle constants like $e$ or $\pi$?

Remember that $e$ and $\pi$ are just numbers, not variables. The derivative of a constant is 0. If you have $e \cdot x$, the derivative is just $e$ (using Constant Multiple Rule), not $0 \cdot x + e \cdot 1$ (though Product Rule still works, it's overkill).

How do I input square roots or fractions?

For square roots, use `sqrt(x)` or `x^(1/2)`. For fractions, use the `/` symbol, like `(x^2+1)/(x-1)`. Make sure to use parentheses to group the numerator and denominator correctly.

Why does the step-by-step solution say 'simplify'?

After applying differentiation rules (like Product or Chain Rule), the raw result is often lengthy. This step attempts to tidy up the expression (e.g., combining like terms). Note that for very complex functions, automatic simplification might not always be as perfect as manual simplification.

Can I use variables other than x?

Currently, our calculator is optimized for the variable $x$. If you have a function with $t$ or $y$, please replace them with $x$ before calculating, or the system might treat them as constants.

Does the order of $f(x)$ and $g(x)$ matter?

No, because multiplication is commutative ($a \cdot b = b \cdot a$) and addition is also commutative. $f'g + fg'$ is the same as $fg' + f'g$. However, keeping a consistent order helps you avoid mistakes.

What is the proof of the Product Rule?

The proof uses the limit definition of the derivative. By adding and subtracting the term $f(x+h)g(x)$ in the numerator of the limit difference quotient, you can split the expression into two parts that converge to $f'(x)g(x)$ and $f(x)g'(x)$.

Derivative Calculator

Product Rule Calculator

The Product Rule is a formula used to find the derivatives of products of two or more functions. For two functions f(x) and g(x), the product rule states that: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

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Step-by-step differentiation with interactive graphs

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Product Rule Calculator

Calculate a Derivative

Introduction

The Formula

How to Use This Calculator

Common Mistakes

Common Examples

Frequently Asked Questions