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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
by logarithmic differentiation.
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Answer

y = (x2 - 5x + 8) (x3 + 7x + 9)
$\Rightarrow\ \log\text{y}=\log(\text{x}^2-5\text{x}+8)+\log(\text{x}^3+7\text{x}+9)$
$\Rightarrow\ \frac{\text{d}}{\text{dx}}\log\text{y}=\frac{\text{d}}{\text{dx}}\log(\text{x}^2-5\text{x}+8)+\frac{\text{d}}{\text{dx}}\log(\text{x}^3+7\text{x}+9)$
$\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{1}{\text{x}^2-5\text{x}+8}\frac{\text{d}}{\text{dx}}(\text{x}^2-5\text{x}+8)+\frac{1}{\text{x}^3+7\text{x}+9}\frac{\text{d}}{\text{dx}}(\text{x}^3+7\text{x}+9)$
$\Rightarrow\ \frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{1}{\text{x}^2-5\text{x}+8}(2\text{x}-5)+\frac{1}{\text{x}^3+7\text{x}+9}(3\text{x}^2+7)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{2\text{x}-5}{\text{x}^2-5\text{x}+8}+\frac{3\text{x}^2+7}{\text{x}^3-7\text{x}+9}\Big]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{(2\text{x}-5)(\text{x}^3+7\text{x}+9)+(3\text{x}^2+7)(\text{x}^2-5\text{x}+8)}{(\text{x}^2-5\text{x}+8)(\text{x}^3+7\text{x}+9)}\Big]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{2\text{x}^4+14\text{x}^2+18\text{x}-5\text{x}^3-35\text{x}-45+3\text{x}^4-15\text{x}^3+24\text{x}^2+7\text{x}^2-35\text{x}+56}{(\text{x}^2-5\text{x}+8)(\text{x}^3+7\text{x}+9)}\Big]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{5\text{x}^4-20\text{x}^3+45\text{x}^2-52\text{x}+11}{(\text{x}^2-5\text{x}+8)(\text{x}^3+7\text{x}+9)}\Big]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=(\text{x}^2-5\text{x}+8)(\text{x}^3+7\text{x}+9)\Big[\frac{5\text{x}^4-20\text{x}^3+45\text{x}^2-52\text{x}+11}{(\text{x}^2-5\text{x}+8)(\text{x}^3+7\text{x}+9)}\Big]\ \text{[From eq.(i)}]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=5\text{x}^4-20\text{x}^3+45\text{x}^2-52\text{x}+11\ \dots\text{(iv)}$
From eq. (ii), (iii) and (iv), we can say that value of $\frac{\text{dy}}{\text{dx}}$ is same obtained by three different methods

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