Find the derivative of the following functions (it is to be under… - Vidyadip

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): $\frac{\text{px}^2\text{+qx+r}}{\text{ax+b}}$

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Answer

Let $\text{f(x)}=\frac{\text{px}^2\text{+qx+r}}{\text{ax+b}}$ By quotient rule, $\text{f}'\text{(x)}=\frac{(\text{ax+b)}\frac{\text{d}}{\text{dx}}(\text{px}^2\text{+qx+r})-(\text{px}^2\text{+qx+r})\frac{\text{d}}{\text{dx}}(\text{ax+b})}{(\text{ax+b})^2}$ $=\frac{(\text{ax+b})(\text{2px+q})-(\text{px}^2\text{qx+r)(a)}}{(\text{ax+b})^2}$ $=\frac{\text{2apx}^2+\text{aqx}+2\text{bpx}+\text{bq}+\text{apx}^2-\text{aqx}-\text{ar}}{(\text{ax+b})^2}$ $=\frac{\text{apx}^2-2\text{bpx+bq}-\text{ar}}{(\text{ax+b})^2}$

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