**Related Pages**

Calculus: Derivatives

Derivative Rules

Calculus: Power Rule

Calculus: Chain Rule

Calculus Lessons

### What Is The Product Rule?

The Product Rule says that the derivative of a product of two functions is the first function
times the derivative of the second function plus the second function times the derivative of the
first function.

The following image gives the product rule for derivatives. Scroll down the page for more examples and solutions.

#### How To Use The Product Rule?

**Example:**

Find f’(x) if f(x) = (6x^{3})(7x^{4})

**Solution:**

Using the Product Rule, we get

**Example:**

Given f(x) = (3x^{2 }– 1)(x^{2} + 5x +2), find the derivative of f(x).

**Solution:**

Using the Product Rule, we get

#### When To Use The Product Rule?

We use the product rule when we need to find the derivative of the product of two functions -
the first function times the derivative of the second, plus the second function times the
derivative of the first.

The product rule is related to the quotient rule, which gives the derivative of the quotient
of two functions, and the chain rule, which gives the derivative of the composite of two functions.

**Example:**

Find the derivative of f(x) = (3x + 5)(2x^{2} - 3)

#### Examples Using The Product Rule And Chain Rule

Find the derivative of

- f(x) = (5x
^{5} - x^{7})(20x^{2} + 3x^{-7})
- f(x) = (10x
^{3} + 5x^{2} - 7)(20x^{8} - 7)
- y = (x
^{2} + 2x)^{5}(3x^{-3} + x^{2})^{-7}

### How To Find Derivatives Using The Product Rule, Chain Rule, And Factoring?

Find the derivative of

f(x) = x^{4}(5x - 1)^{3}

#### How To Use The Product Rule For Derivatives?

**Examples:**

Find the derivative of

- h(x) = (x
^{2})(x^{3} + 4)
- (sin x)(cos x)(x
^{2} + 1)

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