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LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES4 Marks
Question
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\text{a}}{\text{x}^4}-\frac{\text{b}}{\text{x}^2}+\cos\text{x}$
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Answer

Let $\text{f(x)}=\frac{\text{a}}{\text{x}^4}-\frac{\text{b}}{\text{x}^2}+\cos\text{x}$ By quotient rule, $\text{f}'\text{(x)}=\frac{\text{d}}{\text{dx}}\Big(\frac{\text{a}}{\text{x}^4}\Big)-\frac{\text{d}}{\text{dx}}\Big(\frac{\text{b}}{\text{x}^2}\Big)+\frac{\text{d}}{\text{dx}}(\cos\text{x})$ $=\text{a}\frac{\text{d}}{\text{dx}}(\text{x}^{-4})-\text{b}\frac{\text{d}}{\text{dx}}(\text{x}^{-2})+\frac{\text{d}}{\text{dx}}(\cos\text{x})$ $=\text{a}(-4\text{x}^{-5})-\text{b}(-2\text{x}^{-3})+(-\sin\text{x})$ $\Big[\frac{\text{d}}{\text{dx}}(\text{x''})=\text{nx}^{\text{n}-1}\text{and}\frac{\text{d}}{\text{dx}}(\cos\text{x)}=-\sin\text{x}\Big]$ $=\frac{-4\text{a}}{\text{x}^5}+\frac{\text{2b}}{\text{x}^3}-\sin\text{x}$

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