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An online product rule derivative calculator helps you to determine the derivative of a function that is composed of smaller differentiable functions. This calculator uses the product rule of differentiation to simplify your problem precisely. This content is packed with a whole radical information about the product rule.
Read on!
## What Is The Product Rule?

In product rule calculus, we use the multiplication rule of derivatives when two or more functions are getting multiplied.
If we have two functions **f(x)** and **g(x)**, then the product rule states that:
**“ f(x) times the derivative of g(x) plus g(x) times the derivative of f(x)”**
### Formula of Product Rule:

Suppose that we have two functions** f(x)** and** g(x)** that are differentiable. The derivative product rule formula for these functions is as follows:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
Apart from using formula for manual calculations, use online product rule derivative calculator for free to find derivative of two product functions.
### How To Apply Derivative Product Rule?

You can simplify the product of two functions using the basic derivative multiplication rule.
Let us solve a couple of examples.
**Example # 01:**
Differentiate the following function **w.r t x.**
$$ h\left( x \right) = \left( {6{x^2} - x} \right)\left( {1 - 30x} \right) $$
**Solution:**
The given function is:
$$ h\left( x \right) = \left( {6{x^2} - x} \right)\left( {1 - 30x} \right) $$
As we know that the multiplication derivative rule is as follows:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
Here we have separate functions according to the formula as follows:
$$ f(x) = \left( {6{x^2} - x} \right) $$
$$ g(x) = \left( {1 - 30x} \right) $$
Now, calculating the derivative of f(x) w.r.t x:
$$ \frac{d}{d x} f(x) $$
$$ =\frac{d}{d x} \left(6 x^{2} - x\right) $$
$$ \frac{d}{d x} \left(6 x^{2} - x\right) = \left(12x - 1\right) $$
$$ \frac{d}{d x} g(x) $$
$$ =\frac{d}{d x} \left(1 - 30 x\right) $$
$$ \frac{d}{d x} \left(1 - 30 x\right) = -30 $$
(For step-by-step calculation of derivative, click derivative calculator)
Now, according to the multiplication rule for derivatives:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
Putting derivatives in the formula to get the final answer.
$$ \left(6 x^{2} - x\right) (-30) + \left(1 - 30 x\right) \left(12 x - 1\right) $$
$$ - 180 x^{2} + 30 x + \left(1 - 30 x\right) \left(12 x - 1\right) $$
after simplifying , we get:
$$ - 180 x^{2} + 30 x + (1)(12x) - (1)(-1) -(30x)(12x) + (-30x)(-1) $$
$$ - 540 x^{2} + 72 x - 1 $$
So we have:
$$ h\left( x \right) = \left( {6{x^2} - x} \right)\left( {1 - 30x} \right) = - 540 x^{2} + 72 x - 1 $$
Which is the required answer.
Also, our free online product rule derivative calculator evaluates the given functions more accurately and instantly.
**Example # 02:**
Differentiate the following function according to the multiplication rule derivatives w.r t the variable z.
$$ (z^2)^{1/3} *(2z-z^2) $$
**Solution:**
As the product rule is given as:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
We define the functions individually:
$$ f(x) = (z^{2})^{1/3} $$
$$ g(x) = \left(2z - z^{2}\right) $$
Taking derivatives of both the functions:
$$\frac{d}{d z} f(x) $$
$$ =\frac{d}{d z} \sqrt[3]{z^{2}} $$
$$ \frac{d}{d z} \sqrt[3]{z^{2}} = \frac{2 \sqrt[3]{z^{2}}}{3 z} $$
$$ \frac{d}{d z} g(x) $$
$$ =\frac{d}{d z} \left(- z^{2} + 2 z\right) $$
$$ \frac{d}{d z} \left(- z^{2} + 2 z\right) = 2 - 2z $$
(For step-by-step calculation of derivative, click derivative calculator)
Following the product rule for derivatives:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
Putting values for derivatives of each function and simplifying:
$$ \frac{2 \left(5 - 4 z\right) \sqrt[3]{z^{2}}}{3} $$
Which is our required answer.
### How Product Rule Derivative Calculator Works?

To evaluate the derivative of two or more functions that are multiplying, you need to follow a simple guide as follows:
**Input:**
**Output:**
Our free product rule derivatives calculator calculates:
## FAQ’s:

### What is the product rule of exponents?

The product rule of exponents states that:
**“When we multiply the exponential expressions having the same base, we add their exponents”**
$$ \left(a^{m}*a^{n}\right) $$
$$ =a^{m+n} $$
**For example:**
$$ \left(a^{5}*a^{8}\right) $$
$$ =a^{5+8} $$
$$= a^{13} $$
### Can we apply the product rule to 4 terms?

Yes, you can do so. All you need to do is consider derivatives for each new function in the expression and add them to get the final answer.
### How do you express the natural logarithm of zero?

The natural logarithm **(ln)** is defined only for **x>0**. That is why natural log of zero is undefined.
**ln(0) = ∞**
### What is the derivative of log(e)?

As we know that:
**log(e) = 1**. So, we have:
**dy / dx = 0**
The reason is that we know the derivative of any constant term is always zero.
## Conclusion:

The product rule of differentiation has strong applications in the field of calculus and engineering science. Mathematicians make a vast use of free online product rule derivative calculator to differentiate the complex functions at a given point. This calculator helps professionals and students on the same scale to get a generic solution to their problems quickly.
## References:

From the source of wikipedia: Chain rule, Smooth infinitesimal analysis, Quotient rule, Derivatives of inverse functions.
From the source of khan academy: Quotient rule, Differentiate quotients, Quotient rule with table, Differentiating rational functions.
From the source of lumen learning: Derivatives and Rates of Change, The Derivative as a Function, Differentiation Rules, Derivatives of Trigonometric Functions, Implicit Differentiation, Higher Derivatives.

- Enter the given function in the equation menu that is supported by various functions like log, sqrt, ln, sin, cos and tan etc.
- Select the variable w.r.t which you wish to determine the derivative of the function that is given. The available variables are a, b, c, d, x, y, z or n.
- Select limit of differentiation that can't be exceeded than 5.
- Click ‘calculate’

- Overall derivative of the function by following the product rule.
- Simplifies your problem in a proper manner.
- Step by step calculations to better understand the problem structure.

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