Chapter 4

$$ \text { If } s=1+t e^{s}, \text { find } \frac{d^{2} s}{d t^{2}} $$

$$ \text { If } y^{3}+x^{3}-3 a x y=0, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } y=\sin (x+y), \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } e^{x+y}=x y, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } e^{y}+x y=e, \text { find }\left(\frac{d^{2} y}{d x^{2}}\right)_{x=0} $$

$$ \text { If } y=\tan (x+y), \text { find } \frac{d^{3} y}{d x^{3}} $$

$$ \text { If } x^{2}+y^{2}=r^{2}, \text { find }\left(\frac{d^{3} y}{d x^{3}}\right)_{x=0} $$

$$ \text { If } x=\phi(t), y=\psi(t), \text { then find } \frac{d^{2} y}{d x^{2}} \text { . } $$

$$ \text { If } x=a t^{2}, y=2 a t, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } x=a t^{2}, y=b t^{3}, \text { find } \frac{d^{2} y}{d x^{2}} \text { . } $$

$$ \text { If } x=a \cos t, y=a \sin t, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } x=a(t-\sin t), y=a(1-\cos t), \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } x=a \cos ^{2} t, y=a \sin ^{2} t, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } x=\ln t, y=t^{2}-1, \text { find } \frac{d^{2} y}{d x^{2}} $$

$$ \text { If } x=\ln t, y=t^{2}-1, \text { find } \frac{d^{2} y}{d x^{2}} \text { . } $$

$$ \text { If } x=a t \cos t, y=a t \sin t, \text { find }\left(\frac{d^{2} y}{d x^{2}}\right)_{t=0} $$

$$ \text { If } x=a \cos ^{3} t, y=a \sin ^{3} t, \text { find } \frac{d^{3} y}{d x^{3}} $$

$$ \text { If } x=a \cos t, y=b \sin t, \text { find }\left(\frac{d^{3} y}{d x^{3}}\right)_{t=\frac{\pi}{2}} $$

$$ \text { If } y=x^{2} \sin x, \text { find } \frac{d^{25} y}{d x^{25}} \text { . } $$

$$ \text { If } y=e^{x}\left(x^{2}-1\right), \text { find } \frac{d^{24} y}{d x^{24}} $$

$$ \text { If } y=x^{3} \sin x, \text { find } y_{20} $$

$$ \text { If } y=\left(1-x^{2}\right) \cos x, \text { find } y_{2 n} $$

$$ \text { If } y=\frac{3 x+2}{x^{2}-2 x+5}, \text { prove that } y_{n}(0)=\frac{2}{5} n y_{n-1}(0)-\frac{n(n-1)}{5} y_{n-2}(0) \text { for } n \geq 2 \text { . } $$

$$ \text { If } f(x)=3 e^{x^{2}} \text { , then find the value of } f^{\prime}(x)-2 x f(x)+\frac{1}{3} f(0)-f^{\prime}(0) \text { . } $$

$$ \text { If } \sin ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\ln a, \text { then show that } \frac{d y}{d x}=\frac{y}{x} \text { . } $$

$$ \text { If } y=c e^{\frac{x}{(x-a)}}, \text { then show that } \frac{d y}{d x}=-\frac{a y}{(x-a)^{2}} \text { . } $$

$$ \text { If } x^{y}=e^{x-y}, \text { then show that } \frac{d y}{d x}=\frac{\ln x}{(1+\ln x)^{2}} \text { . } $$

$$ \text { If } y=e^{x+e^{x+e^{x} \cdots}}, \text { prove that } \frac{d y}{d x}=\frac{y}{1-y} $$

$$ \text { If } y=x^{x^{x \ldots}}, \text { prove that } \frac{d y}{d x}=\frac{y^{2}}{(1-y \ln x) x} $$

$$ \text { If } y=a^{x^{x^{x-\infty}}}, \text { show that } \frac{d y}{d x}=\frac{y^{2} \ln y}{x(1-y \ln x \ln y)} \text { . } $$

$$ \text { If } x=\sec \theta-\cos \theta, y=\sec ^{n} \theta-\cos ^{n} \theta \text { then prove that }\left(x^{2}+4\right)\left(y^{\prime}\right)^{2}=n^{2}\left(y^{2}+4\right) \text { . } $$

$$ \text { If } y=a+b x^{2}, \text { where } a \text { and } b \text { are arbitrary constants, then prove that } x \frac{d^{2} y}{d x^{2}}=\frac{d y}{d x} \text { . } $$