GATE 2023 BT – Question 59

Correct answer: -2

Explanation

**Quotient rule.** For $f=\dfrac{N}{D}$ with $N=\sin x+\cos x$ and $D=\sin x-\cos x$:
$$f^\prime=\frac{N^\prime D-N D^\prime}{D^2}.$$

**Derivatives.**
$$N^\prime=\cos x-\sin x,\qquad D^\prime=\cos x+\sin x .$$

**At $x=0$.**
$$N=1,\ D=-1,\ N^\prime=1,\ D^\prime=1 .$$

$$f^\prime(0)=\frac{(1)(-1)-(1)(1)}{(-1)^2}=\frac{-2}{1}=\mathbf{-2}.$$