Product Rule Formula

Product Rule Formula

A product rule in Calculus is a method of finding the derivative or derivative of a given function with respect to the product of two differentiable functions. That is, students can apply the product rule or Leibniz’s law to find the derivative of a function of form f(x) g(x) such that both f(x) and g(x) are differentiable. The product rule is the rule which directly follows the concept of limits and derivatives in differentiation and students can understand the Product Rule Formula and its proof in detail with a solved example from the website of Extramarks.

What Is The Product Rule?

The product rule in Calculus is the method used to find the derivative of any function given in the form of the product obtained by multiplying any two differentiable functions. The product rule states that the derivative of the product of two differentiable functions is equal to the sum of the product of the differentiation of the second function and the first function plus the product of the differentiation of the first function and the second function. That is, given a function of form f(x) g(x), students can find the derivative of that function using the product-law derivative as follows:

f(x) g(x) = [g(x) × f`(x) + f(x) × g'(x)]

Derivation Of Product Rule Formula

The Product Rule Formula is a significant theme in Mathematics. Students can compute derivatives or evaluate the derivative of the product of two functions using the Calculus Product Rule Formula. 

Also f(x) = product of differentiable functions u(x) and v(x)

u(x), v(x) = differentiable function

u'(x) = derivative of function u(x)

v'(x) = derivative of function v(x)

Derivation of the Product Rule Formula is also an important theme. In the previous section, students learned about the product formula for finding the derivative of the product of two differentiable functions. For any two functions, the product rule can be specified in Lagrangian notation as

 (u v)’ = u’ v + u v’

or in Leibniz notation

Students should look at the proof of the multiplication formula. 

• Differentiation rule

If a function is differentiable on all points [a,b], then the function is differentiable on the interval [a,b]. The sum, difference, product, and composition of differentiable functions are differentiable wherever they are defined, and the quotient of two differentiable functions is differentiable wherever they are defined. The differentiation rule is

•  Sum rule: dy/dx = du/dx ± dv/dx if y = u(x) ± v(x). Product rule: dy/dx = if y = u(x) × v(x) u.dv/dx + v.du/dx
• Quotient law: if y = u(x) ÷ v(x) then dy/dx = (v.du/dx- u.dv/dx)/ v2
• Chain rule: Let y = f(u) be a function of u, and if u=g(x) such that y = f(g(x), then d/dx(f(g(x))= f` (g(x))g'(x)
• Constant rule: y = k f(x), for k ≠ 0, d/dx(k(f(x))) = k d/dx f(x).

Product Rule Formula Proof Using First Principle

A Product Rule Formula is a formula used to find the derivative of the product of two or more functions. Suppose u(x) and v(x) are differentiable functions. Thus, the product of the functions u(x)v(x) is also differentiable, written as multiplying the derivative of the first function to compute the function gives the product rule. Here students take the u constant in the first term and the v constant in the second term.

Students can compute derivatives, or use the Product Rule Formula to compute the derivative of the product of two functions in which f(x) = product of differentiable functions u(x) and v(x)

u(x), v(x) = differentiable function

u'(x) = derivative of function u(x)

v'(x) = derivative of function v(x)

Students are advised to learn the formulas form Extramarks website as it is a student-friendly website which provides the best solutions for every problem related to examinations.

Product Rule Formula Proof Using Chain Rule

By considering the product rule as a special case of the chain rule, the chain rule formula can be used to derive the Calculus Product Rule Formula. Let f(x) be a differentiable function of h(x) = f(x) g(x).

i.e.

i.e.

X

(fg) = [δ(fg)/δf][df/dx] + [δ(fg)/δg][dg/dx] = g(df/dx) + f(dg/dx)

So proven.

Products with more than two features have their own product rules. The product rule can be generalised to products of three or more factors using the same Product Rule Formula. For example, for three functions u(x), v(x), and w(x), the product is given as u(x)v(x)w(x).

It is crucial to understand how students use the Product Rule Formula for differentiation. To find the derivative of a function of the form h(x) = f(x)g(x), both f(x) and g(x) must be differentiable functions. The following procedure can be applied to find the derivative of a function h(x) = f(x)g(x) that is differentiable using the product rule.

Step 1: Write down the values ​​of f(x) and g(x). Step 2: Find the values ​​of f`(x) and g'(x), h'(x) =

i.e.

i.e.

X

f(x) g(x) = [g(x) × f'(x) + f(x) × g'(x)]

Students must take a look at the following example to better understand the product rule.

Product Rule For Product Of More Than Two Functions

A Product Rule Formula in Calculus is a method of finding the derivative or derivative of a given function with respect to the product of two differentiable functions. That is, students can apply the Product Rule Formula or Leibniz’s law to find the derivative of a function of form f(x) g(x) such that both f(x) and g(x) are differentiable increases. The Product Rule Formula directly follows the concept of limits and derivatives in differentiation. In the next section, students would understand the Product Rule Formula and its proof in detail with a solved example.

It is important to comprehend what are the product rules. The Product Rule Formula in Calculus is the method used to find the derivative of any function given in the form of the product obtained by multiplying any two differentiable functions. The Product Rule Formula states that the derivative of the product of two differentiable functions is equal to the sum of the product of the second function by the differentiation of the first function and the product of the differentiation of the first function by the second function about it. That is, given a function of form f(x) g(x), students can find the derivative of that function using the product-law derivative as follows:

i.e.

X

f(x) g(x) = [g(x) × f`(x) + f(x) × g'(x)]

How To Apply Product Rule In Differentiation?

The process of finding the derivative of a function is called Calculus and derivative is the rate of change of a function with respect to another quantity. The Laws of Differential Calculus were established by Sir Isaac Newton. The principle of limits and derivatives is used in many scientific fields. Differentiation and integration form the main concepts of analysis. Students must learn differentiating techniques to find derivatives of algebraic, trigonometric and exponential functions.

It is significant to understand what is differentiation. Derivative means the rate of change of one quantity with respect to another quantity. Velocity is calculated as the rate of change of distance over time. At no point in time is this velocity the same as the calculated average. Velocity is the same as slope, it is just the instantaneous rate of change in distance over a period of time.The ratio of a small change in one quantity to a small change in another quantity that depends on the first quantity is called the derivative. One of the most important concepts in Calculus mainly focuses on the derivatives of functions. The maxima or minima of functions, the velocity and acceleration of moving objects, and the tangents of curves are determined by differentiation. If y = f(x) is differentiable, the derivative is expressed as f`(x) or dy/dx.

Derivative definition

The geometric meaning of the derivative of y = f(x) is the slope of the tangent to the curve y = f(x) at ( x, f(x)). The first principle of differentiation is to use bounds to compute the derivative of a function. The curve function is y = f(x). Let’s take a point P with coordinates (x, f(x)) on the curve. Take another point Q on the curve with coordinates (x+h, f(x+h)). where PQ is the secant of the curve. The slope of the curve at a point is the slope of the tangent line at that point.

Examples On Product Rule

Example: Use the Product Rule Formula f(x) = x log x to find f`(x) in the following function f(x). 

Solution:  

where f(x) = x log x applies

u(x) = x

v(x) = log x

 ⇒u'(x)=1

⇒v'(x) = 1/x

 ⇒f'(x) = [v(x)u'(x) + u(x)v'(x)]

⇒f'(x) = [log x 1 + x (1/x)]

⇒ f'(x) = log x + 1

 Answer: The derivative of x log x using the Product Rule Formula is log x + 1.

Practice Questions On Product Rule

1. What are the product differentiation rules in Calculus? 

Something is one of the derivative rules used to find the derivation of the function of the P (x) = f (x) g (x) format and the derivative of a function P(x) is denoted by P`(x). A function P(x) is said to be differentiable if a derivative of it exists. In other words, a differentiable function is a function for which there exists a derivative. A function P(x) is differentiable at the point x = a if the following limit exists

2. How can students find derivatives using the product rule? 

The derivative of the product of two differentiable functions can be calculated in Analysis using the product rule. To derive a function of the form f(x) = u(x)v(x), students need to use the Product Rule Formula and it should be noted that the Product Rule Formula is given by:

f'(x) = [u(x)v(x)]’ = [u'(x) × v(x) + u(x) × v'(x)]

where f'(x), u'(x), and v'(x) are the derivatives of the functions f(x), v(x), and u(x).

3. What is a Product Rule Formula expression? 

The Product Rule Formula derivative formula is a Calculus rule used to find the derivative of the product of two or more functions. Assuming that the two functions u(x) and v(x) are differentiable, students can apply the Product Rule Formula to find (d/dx)[u(x)v(x)] and they can do it.

 f'(x) = [u(x)v(x)]’ = [u'(x) × v(x) + u(x) × v'(x)]

4. How to derive the Product Rule Formula?

 The Product Rule Formula can be derived using a variety of methods. They are given as

using derivatives and limit properties or first principles and they can also apply the chain rule.

5. How are product rules used for differentiation? 

Students can use the Product Rule Formula to find the derivative of a function f(x) of the form u(x)v(x). The derivative of this function using the Product Rule Formula is given by f'(x) = [u(x)v(x)]’ = [u'(x) × v(x) + u(x) × v'(x)]