Differentiate the following functions with respect to x: 4x+5 3x+… - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{\text{4x}+5\sin\text{x}}{\text{3x}+7\cos\text{x}}$

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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{4x}+5\sin\text{x}}{\text{3x}+7\cos\text{x}}\Big)$
Using quotient rule, we get
$\frac{(\text{3x}+7\cos\text{x})\frac{\text{d}}{\text{dx}}(\text{4x}+5\sin\text{x})-(4\text{x}+5\sin\text{x})\frac{\text{d}}{\text{dx}}(\text{3x}+7\cos\text{x})}{(\text{3x}+7\cos\text{x})^2}$
$=\frac{(\text{3x}+7\cos\text{x})(4+5\cos\text{x})-(4\text{x}+5\sin\text{x})(3+7(-\sin\text{x}))}{(\text{3x}+7\cos\text{x})^2}$
$=\frac{12\text{x}+28\cos\text{x}+\text{15x}\cos\text{x}+13\cos^2\text{x}-\text{12x}-15\sin\text{x}+\text{28x}\sin\text{x}+25\sin^2\text{x}}{(\text{3x}+7\cos\text{x})^2}$
$=\frac{\text{15x}\cos\text{x}+\text{28x}\sin\text{x}+28\cos\text{x}-15\sin\text{x}+35(\sin^2\text{x}+\cos^2\text{x})}{(\text{3x}+7\cos\text{x})^2}$
$\therefore\frac{\text{15x}\cos\text{x}+\text{28x}\sin\text{x}+28\cos\text{x}-15\sin\text{x}+35}{(\text{3x}+7\cos\text{x})^2}$

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