UV Differentiation Formula
UV differentiation formula helps to find the differentiation of the product of two functions. The product rule is one of the derivative rules that we use to find the derivative of two or more functions. The uv differentiation formula has various applications in partial differentiation and in integration.
Let us try to know the uv differentiation formula, the different methods to prove this formula, its applications, and the examples of uv formula of differentiation.
What Is UV DIfferentiation Formula?
UV Differentiation formula is an important formula in differentiation. The uv formula in differentiation is the sum of the differentiation of the first function multiplied with the second function, and the differentiation of the second function multiplied with the first function. The uv differentiation formula for two functions is as follows.
(uv)' = u'.v + u.v'
Also the two functions are often represented as f(x), and g(x), and the differentiation of the product of these two functions is d/dx (f(x).g(x)) = g(x).d/dx f(x) + f(x). d/dx g(x). Similar to this uv formula in differentiation, we have a uv formula for integration.
Proof of UV Differentiation Formula
The UV differentiation formula can be proved through various methods. Some of the important methods of providing the uv differentiation formula is through the method of first principle, through infinitesimal analysis, and through the use of logarithmic functions. let us check the detailed proof of each of these methods.
Proof  From First Principle
Product rule formula is a formula that we use to find the derivative of two or more functions. Suppose two functions, \(f(x)\) and \(g(x)\) are differentiable. The proof of the product is as follows.
\[\begin{align}\frac{{d\left\{ {f\left( x \right)g\left( x \right)} \right\}}}{{dx}}&=\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right)g\left( {x + h} \right)  f\left( x \right)g\left( x \right)}}{h}\\&=\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right)g\left( {x + h} \right)  f\left( {x + h} \right) g\left( {x} \right) + f\left( {x + h} \right) g\left( {x} \right)  f\left( x \right)g\left( x \right)}}{h}\\&=\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right)\left\{ {g\left( {x + h} \right)  g\left( x \right)} \right\} + g\left( x \right)\left\{ {f\left( {x + h} \right)  f\left( x \right)} \right\}}}{h}\\& = \mathop {\lim }\limits_{h \to 0} f\left( {x + h} \right)\mathop {\lim }\limits_{h \to 0} \frac{{g\left( {x + h} \right)  g\left( x \right)}}{h} + g\left( x \right)\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right)  f\left( x \right)}}{h}\\&=f\left( x \right)\frac{{d\left( {g\left( x \right)} \right)}}{{dx}} + g\left(x \right)\frac{{d\left( {f\left( x \right)} \right)}}{{dx}}\end{align}\]
Thus, [f(x).g(x)]' = f'(x).g(x) + g'(x).f(x). Further we can replace f(x) = u, and g(x) = v, to obtain the final expression.
(uv)' = u'.v + v'.u
Proof  Infinitesimal Analysis
The basic application of derivative is in the use of it to find the errors in quantities being measures. Let us consider the two functions as two quantities u and v respectively. The total error in the product of these quantities is equal to the difference of the product of the measured quantities and the product of the actual quantities.
d (uv) = (u + dv)(v + dv)  uv
=uv + udv + vdu + du.dv  uv
= u.dv + v.du + du.dv
= u.dv + v.du
Proof  Using Logarithmic Formula
The proof of uv differential can also be derived using logarithms. First, we apply logarithms to the product of the functions uv, and then we apply derivatives to obtain the final uv formula of differentiation. Here we use the formula of logarithm  logab = loga + logb, and the formula of differentiation of this logarithmic function  d/dx.log f(x) = f'(x)/f(x). Let us check the below steps to prove the uv formula of differentiation, in the below steps.
log(u.v) = logu + logv
Applying derivatives on both sides we have
d/dx.(log(uv)) = d/dx.logu + d/dx.logv
(uv)'/(uv) = u'/u + v'/v
(uv)' = (uv)[u'/u + v'/v]
(uv)' = uv.u'/u + uv.v'/v
(uv)' = u'v + v'u
Application of UV Differentiation Formula
The uv differentiation formula has numerous applications in calculus. Some of the important applications are as follows.
 The uv formula of differentiation can also be used to find the differentiation of the product of three or more functions.
 The uv differentiation has helped in the concept of partial derivatives of two or more functions.
 The integration of the product of two functions has also been derived with the help of this uv differentiation formula.
Related Topics
The following topics are helpful for a clear understanding of uv differentiation formula.
Examples on UV Differentiation Formula

Example 1: Find the derivative of \(x^2Tan{x}\) using the. uv formula of differentiation
Solution:
Let \(f(x) = x^2\) and \(g(x)=Tan{x}\).
\(\begin{align}\ \dfrac{d}{dx}(x^2.Tanx) &=(\frac{d}{dx}x^2).Tanx + (\frac{d}{dx}Tanx).x^2\\&=2x.Tanx + Sec^2x.x^2\\&=2x.Tanx + x^2.Sec^2x\\&=x(2Tanx + xSec^2x)\end{align}\)
Hence, the differentiation of \(x^2Tan{x}\) is \(x(2Tanx + xSec^2x)\).

Example 2: Differentiate \(e^x\sqrt x\) using the uv differentiation formula.
Solution:
Let \(f(x) = e^x\) and \(g(x)=\sqrt x\).
\(\begin{align}\ \dfrac{d}{dx}(e^x.\sqrt x) &=(\frac{d}{dx}e^x).\sqrt x + (\frac{d}{dx}\sqrt x).e^x\\&=e^x.\sqrt x + \frac{1}{2\sqrt x}.e^x\\&=e^x(\sqrt x + \frac{1}{2\sqrt x})\end{align}\)
Hence, the differentiation of \(e^x \sqrt x\) is \(e^x(\sqrt x + \frac{1}{2\sqrt x})\)
FAQs on UV Differentiation Formula
What Is UV Differentiation Formula?
The uv differentiation formula is (uv)' = u'v + v'u. This is used to find the differentiation of the product of two functions.
How Do We Use UV Differentiation Formula?
The uv differentiation formula is used to find the differentiation of the product of two functions. The differentiation of the product of two functions is equal to the sum of the differentiation of the first function multiplied with the second function, and the differentiation of the second function multiplied with the first function. For two functions u and v the uv differentiation formula is (u.v)' = u'v + v'u.
What Are The Uses of UV Differentiation Formula?
The uv differentiation formula can be used to find the differentiation of the product of two functions. Also, this formula can be used for the differentiation of the product of three or more functions. The concept of partial derivatives has also been derived from this differentiation formula.
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