Differentiate \((x^5-5x+8)(x^3+7x+9)\) in three ways mentioned below
(i) By using product rule
(ii) By expanding the product to obtain a single polynomial.
(iii) By logarithmic differentiation.
Do they all give the same answer?
(i) By using product rule
(ii) By expanding the product to obtain a single polynomial.
(iii) By logarithmic differentiation.
Do they all give the same answer?
Solution and Explanation
let \(x^5-5x+8=u\) and \(x^3+7x+9=v\)
\(∴y=uv\)
\(⇒\frac{dy}{dx}=\frac{du}{dx}.v+u.\frac{dv}{dx}\) [by product rule]
\(⇒\frac{dy}{dx}=\frac{d}{dx}(x^5-5x+8).(x^3+7x+9)+(x^5-5x+8).\frac{d}{dx}(x^3+7x+9)\)
\(⇒\frac{dy}{dx}=(2x-5)(x^3+7x+9)+(x^5-5x+8)(3x^2+7)\)
\(⇒\frac{dy}{dx}=2x(x^3+7x+9)-5(x^3+7x+9)+x^2(3x^2+7)-5x(3x^2+7)+8(3x^2+7)\)
\(⇒\frac{dy}{dx}=(2x^4+14x^2+18x)-5x^3-35x-45+(3x^4+7x^2)-15x^3-35x+24x^2+56\)
\(∴\frac{dy}{dx}=5x^4-20x^3+45x^2-52x+11\)
(ii)\(y=(x^5-5x+8)(x^3+7x+9)\)
\(=x^2(x^3+7x+9)-5x(x^3+7x+9)+8(x^3+7x+9)\)
\(=x^5+7x^3+9x^2-5x^4-35x^2-45x+8x^3+56x+72\)
\(=x^5-5x^4+15x^3-26x^2+11x+72\)
\(∴\frac{dy}{dx}=\frac{d}{dx}(x^5-5x^4+15x^3-26x^2+11x+72)\)
\(=\frac{d}{dx}(x^5)-5\frac{d}{dx}(x^4)+15\frac{d}{dx}(x^3)-26\frac{d}{dx}(x^2)+11\frac{d}{dx}(x)+\frac{d}{dx}(72)\)
\(=5x^4-5\times4x^3+15\times3x^2-26\times2x+11\times1+0\)
\(=5x^4-20x^3+45x^2-52x+11\)
(iii)\(y=(x^2-5x+8)(x^3+7x+9)\)
Taking logarithm on both the sides,we obtain
\(logy=log(x^2-5x+8)+log(x^3+7x+9)\)
Differentiating both sides with respect to \(x\),we obtain
\(\frac{1}{y}\frac{dy}{dx}=\frac{d}{dx}log(x^2-5x+8)+\frac{d}{dx}log(x^3+7x+9)\)
\(⇒\frac{1}{y}\frac{dy}{dx}=\frac{1}{x^5-5x+8}.\frac{d}{dx}(x^2-5x+8)+\frac{1}{x^3+7x+9}.\frac{d}{dx}(x^3+7x+9)\)
\(⇒\frac{dy}{dx}=y[\frac{1}{x^2-5x+8}.(2x-5)+\frac{1}{x^3+7x+9}.(3x^2+7)]\)
\(⇒\frac{dy}{dx}=(x^2-5x+8)(x^3+7x+9)[\frac{2x-5}{(x^2-5x+8)}+\frac{3x^2+7}{(x^3+7x+9)}]\)
\(⇒\frac{dy}{dx}=(x^2-5x+8)(x^3+7x+9)[\frac{(2x-5)(x^3+7x+9)+(3x^2+7)(x^2-5x+8)}{(x^2-5x+8)(x^3+7x+9)}]\)
\(⇒\frac{dy}{dx}=2x(x^3+7x+9)-5(x^3+7x+9)+3x^2(x^2-5x+8)+7(x^2-5x+8)\)
\(⇒\frac{dy}{dx}=(2x^4+14x^2+18x)-5x^3-35x-45+(3x^4-15x^3+24x^2)+(7x^2-35x+56)\)
\(⇒\frac{dy}{dx}=5x^4-20x^3+45x^2-52x+11\)
From the above three observations, it can be concluded that all the results of dy/dx are same
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Concepts Used:
Logarithmic Differentiation
Logarithmic differentiation is a method to find the derivatives of some complicated functions, using logarithms. There are cases in which differentiating the logarithm of a given function is simpler as compared to differentiating the function itself. By the proper usage of properties of logarithms and chain rule finding, the derivatives become easy. This concept is applicable to nearly all the non-zero functions which are differentiable in nature.
Therefore, in calculus, the differentiation of some complex functions is done by taking logarithms and then the logarithmic derivative is utilized to solve such a function.