Learn all About the Product Rule

Calculus employs the product rule to differentiate functions. The product rule is employed when a given function is the product of two or more other functions. If the issues are a combination of two or more functions, the product rule can be used to find their derivatives. Simply put, the term “product” refers to the combination of two functions that are multiplied together.

This rule, which was discovered by Gottfried Leibniz, allows us to calculate derivatives that we don’t want (or are unable to) multiply rapidly.

With another way of putting it, the product rule allows us to find the derivative of two differentiable functions that are multiplied together by combining our knowledge of both the power rule for derivatives and the sum and difference rule for derivatives.

It can be expressed as follows in simple terms: The second times the derivative of the first multiplied by the derivative of the first multiplied by the second times its own derivative equals the derivative of the second times its own derivative.

Definition  

The product rule is a generic rule that can be applied to issues that fall under the differentiation category and involve the multiplication of one function by another. Taking the derivative of the product of two differentiable functions is identical to the addition of the first function multiplied by the derivative of the second, and the addition of the second function multiplied by the derivative of the first. The function could be an exponential function, a logarithmic function, or something else.  In accordance with the rule of the power of a product, we can simplify a power of power when we multiply exponents while keeping the same base constant.

The formula of product rule 

Suppose we have a function y = uv, in which the functions of x are denoted by u and v, respectively. In this case, we can easily find out the derivative of y with respect to x by applying the product rule, which may be represented as:

In the equation, (dy/dx) = u (dv/dx) + v (du/dx).

The product rule for derivatives, often known as the product rule of differentiation, is the formula presented above.

In the first term, we regarded u to be a constant, and in the second term, we considered v to be a constant as well.

Zero product rule 

According to the zero product rule, the sum of two non-zero numbers is zero only if one of them is zero. If a and b are two numbers, then ab = 0 only if either a or b is equal to zero. Otherwise, ab = 0.

If (x-1)x = 0, then either x – 1 = 0 or x = 0 is true; otherwise, neither is true.

Essentially, it indicates that if x–1 = 0, then x = 1.

The values of x are between 0 and 1. They are sometimes referred to as the equation’s roots. In most cases, this method is used to discover the roots of equations, and it is effective if one side of the equation is zero.

Example of product rule 

Reduce the complexity of the expression to: y= x2×x5

Solution:

Given the equation: y=x2 × x5

We are aware of the fact that the product rule for the exponent is

xn × xm = xn+m is the product of xn and xm.

It is possible to write it as follows using the product rule:

y = x2×x5 = x2+5 = y = x2 × x5

y = x 7  is a mathematical equation.

As a result, the simplified form of the expression y= x2 × x5 is represented by the symbol x7.

Instantaneous Rate Of Change 

In order to illustrate this, let’s look at two more examples.

If we are asked to find the derivative of a function h(x) at the value of 1, as shown in the example below, we can use the following formula:

After determining how to calculate the derivative, we must plug in the number 1 to determine the instantaneous rate of change, which is the instantaneous rate of change, which is the instantaneous rate of change.

h(x)=(5-7x²)(2x³-9) 

h'(1)=?

h(x)=(5-7x2)(2x3-9)

h'(x)=(5-7x2)[6x2]+(2x3-9)[-14x]

h'(x)= (30x2 – 42x4) + (126x – 28 x4 )

H'(x)= -70x4 + 30x2 +126x

h'(1)= -70(1)4 +30(1)2 +126(1)

h'(1)= 86

In order to solve this problem, we simply rewrote the first function and multiplied it by the derivative of the second function, and then added the product of the second function and its derivative to arrive at the solution. Finally, we discovered that the derivative at the point x = 1 is equal to 86.

First, simplify, and then, using the power rule, determine the derivative.

Alternatively, you can use the product rule and then simplify it.

Conclusion 

Calculus employs the product rule to differentiate functions. The product rule is employed when a given function is the product of two or more other functions. The product rule is a generic rule that can be applied to issues that fall under the differentiation category and involve the multiplication of one function by another. Taking the derivative of the product of two differentiable functions is identical to the addition of the first function multiplied by the derivative of the second, and the addition of the second function multiplied by the derivative of the first.