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Product Rule Formula

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  • Last Updated : 08 Mar, 2022
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Calculus is a discipline of mathematics that deals with continuous change and is one of the most significant branches of mathematics. Calculus is built around two key concepts: derivatives and integrals. The rate of change of a function is measured by its derivative, whereas the area under the curve of the function is measured by its integral. The integral collects the discrete values of a function across a range of values, whereas the derivative offers the explanation of the function at a given point.

Product Rule

In calculus, the product rule is a technique for determining the derivative of any function that is presented in the form of a product produced by multiplying two differentiable functions. The derivative of a product of two differentiable functions is equal to the sum of the product of the second function with differentiation of the first function and the product of the first function with differentiation of the second function, according to the product rule.

If we have a function of type f(x)⋅g(x), we can use the product rule derivative to obtain the derivative of that function. The formula for the product rule is as follows:

d(u(x).v(x))​/dx = [v(x)×u′(x)+u(x)×v′(x)]

where,

u(x) and v(x) are differentiable functions in R.

u'(x) and v'(x) are the derivatives of functions u(x) and v(x) respectively.

Derivation

Suppose a function f(x) = u(x)⋅v(x) is differentiable at x. We will prove the product rule formula using the definition of derivative or limits.

f'(x)=\lim_{\Delta x\to0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

\lim_{\Delta x\to0} \frac{u(x+\Delta x)⋅v(x+\Delta x)-u(x)⋅v(x)}{\Delta x}

\lim_{\Delta x\to0} \frac{u(x+\Delta x)⋅v(x+\Delta x)-u(x)⋅v(x+\Delta x)+u(x)⋅v(x+\Delta x)-u(x)⋅v(x)}{\Delta x}

\lim_{\Delta x\to0} \frac{(u(x+\Delta x)-u(x))⋅v(x+\Delta x)+u(x)⋅(v(x+\Delta x)-v(x))}{\Delta x}

\lim_{\Delta x\to0} \frac{(u(x+\Delta x)-u(x))⋅v(x+\Delta x)}{\Delta x}+\lim_{\Delta x\to0}\frac{u(x)⋅(v(x+\Delta x)-v(x))}{\Delta x}

v(x)\lim_{\Delta x\to0} \frac{u(x+\Delta x)-u(x)}{\Delta x}+u(x)\lim_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}

Put \lim_{\Delta x\to0} \frac{u(x+\Delta x)-u(x)}{\Delta x}=u'(x)   and \lim_{\Delta x\to0}\frac{v(x+\Delta x)-v(x)}{\Delta x}=v'(x)

= v(x) × u'(x) + u(x) × v'(x)

This derives the formula for product rule.

Sample Problems

Question 1. Find the derivative of the function f(x) = x sin x using product rule.

Solution:

We have, f(x) = x sin x. Here, u(x) = x and v(x) = sin x.

So, u'(x) = 1 and v'(x) = cos x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sin x (1) + x (cos x)

= sin x + x cos x

Question 2. Find the derivative of the function f(x) = x log x using product rule.

Solution:

We have, f(x) = x log x. Here, u(x) = x and v(x) = log x.

So, u'(x) = 1 and v'(x) = 1/x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= log x (1) + x (1/x)

= log x + 1

Question 3. Find the derivative of the function f(x) = x2 cos x using product rule.

Solution:

We have, f(x) = x2 cos x. Here, u(x) = x2 and v(x) = cos x.

So, u'(x) = 2x and v'(x) = -sin x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= cos x (2x) + x2(-sin x)

= 2x cos x – x2 sin x

Question 4. Find the derivative of the function f(x) = sin x log x using product rule.

Solution:

We have, f(x) = sin x log x. Here, u(x) = sin x and v(x) = log x.

So, u'(x) = cos x and v'(x) = 1/x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= log x (cos x) + sin x (1/x)

= log x cos x + sin x/ x

Question 5. Find the derivative of the function f(x) = tan x sec x using product rule.

Solution:

We have, f(x) = tan x sec x. Here, u(x) = tan x and v(x) = sec x.

So, u'(x) = sec2 x and v'(x) = sec x tan x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sec x (sec2 x) + tan x (sec x tan x)

= sec x (sec2 x + tan2 x)

= sec x (2sec2 x – 1)

Question 6. Find the derivative of the function f(x) = (x – 3) sin x using product rule.

Solution:

We have, f(x) = (x – 3) cos x. Here, u(x) = x – 3 and v(x) = sin x.

So, u'(x) = 1 and v'(x) = cos x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sin x (1) + (x – 3) (cos x)

= sin x + x cos x – 3 cos x

Question 7. Find the derivative of the function f(x) = x sec x using product rule.

Solution:

We have, f(x) = x sec x. Here, u(x) = x and v(x) = sec x.

So, u'(x) = 1 and v'(x) = sec x tan x

Using product rule we have,

f'(x) = v(x)u'(x) + u(x)v'(x)

= sec x (1) + x (sec x tan x)

= sec x (1 + x tan x)


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