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Q.
If $\ln (3 \sin x-4 \cos x+7+5 y)=(\sin x) y$, then $y^{\prime}(\pi)$ is equal to
Continuity and Differentiability
Solution:
We have $\ln (3 \sin x-4 \cos x+7+5 y)=(\sin x) y$......(1)
Put $x=\pi$ in equation (1), we get $\ln (11+5 y)=0 \Rightarrow 11+5 y=1 \Rightarrow 5 y=-10$
$\therefore y=-2$. So, $(\pi,-2)$ lies on the given curve.
Now on differentiating both the sides of equation (1) w.r.t. $x$, we get
$\frac{1}{(3 \sin x-4 \cos x+7+5 y)} \times\left(3 \cos x+4 \sin x+5 \frac{d y}{d x}\right)=(\sin x) \frac{d y}{d x}+(\cos x) y$
As $(\pi,-2)$ satisfy it, we get $1 \times\left(-3+0+5 \frac{ dy }{ dx }\right)=2$
$\frac{ dy }{ dx }=\frac{5}{5}=1 $