Differentiate the following functions. 3x+4 5x^ 2 -7x+9 - Vidyadip

Question

Differentiate the following functions.
$\frac{3\text{x}+4}{5\text{x}^{2}-7\text{x}+9}$

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Answer

$\frac{\text{d}}{\text{dx}}\Big(\frac{3\text{x}+4}{5\text{x}^{2}-7\text{x}+9}\Big)$
$=\frac{(5\text{x}^{2}-7\text{x}+9)\frac{\text{d}}{\text{dx}}(3\text{x}+4)-(3\text{x}+4).\frac{\text{d}}{\text{dx}}(5\text{x}^{2}-7\text{x}+9)}{(5\text{x}^{2}-7\text{x}+9)^{2}}$
$=\frac{(5\text{x}^{2}-7\text{x}+9)(3)-(3\text{x}+4)(10\text{x}-7)}{(5\text{x}^{2}-7\text{x}+9)^{2}}$
$=\frac{15\text{x}^{2}-21\text{x}+27-30\text{x}^{2}+21\text{x}-40\text{x}+28}{(5\text{x}^{2}-7\text{x}+9)^{2}}$
$=\frac{-15\text{x}^{2}-40\text{x}+55}{(5\text{x}^{2}-7\text{x}+9)^{2}}$
$=\frac{55-40-15\text{x}^{2}}{(5\text{x}^{2}-7\text{x}+9)^{2}}$
Hence, the required answer is $\frac{55-40-15\text{x}^{2}}{(5\text{x}^{2}-7\text{x}+9)^{2}}.$

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