Question

Find the derivative of $\frac{4x+5\sin x}{3x+7\cos x}$

Answer 1

Let $f\left(x\right)=\frac{4x+5\sin x}{3x+7\cos x}$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(\frac{4x+5\sin x}{3x+7\cos x}\right)$
$\Rightarrow f^{\prime }\left(x\right)=\frac{(3x+7\cos x)\frac{d}{dx}(4x+5\sin x)-(4x+5\sin x)\frac{d}{dx}(3x+7\cos x)}{(3x+7\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{(3x+7\cos x)(4+5\cos x)-(4x+5\sin x)(3-7\sin x)}{(3x+7\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{4(3x+7\cos x)+5\cos x(3x+7\cos x)-\left[3\right(4x+5\sin x)-7\sin x(4x+5\sin x\left)\right]}{(3x+7\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{12x+28\cos x+15x\cos x+35{\cos }^{2}x-(12x+15\sin x-28x\sin x-35{\sin }^{2}x)}{(3x+7\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{28\cos x+28x\sin x+15x\cos x-15\sin x+35{\cos }^{2}x+35{\sin }^{2}x}{(3x+7\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{28(\cos x+x\sin x)+15(x\cos x-\sin x)+35({\sin }^{2}x+{\cos }^{2}x)}{(3x+7\cos x{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{28(\cos x+x\sin x)+15(x\cos x-\sin x)+35}{(3x+7\cos x{)}^2}$