91Ó°ÊÓ

Find \(\frac{d^{2} y}{d x^{2}}\), if (i) \(x=a t^{2}, y=2 a t\) (ii) \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\) (iii) \(x=a \cos \theta, y=b \sin \theta\)

\text { If } x=a \sec \theta, y=b \tan \theta \text { prove that } \frac{d^{2} y}{d x^{2}}=-\frac{b^{4}}{a^{2} y^{3}} .

\begin{aligned} &\text { If } x=a(1-\cos \theta), y=a(\theta+\sin \theta) \text {, prove that }\\\ &\frac{d^{2} y}{d x^{2}}=-\frac{1}{a} \text { at } \theta=\frac{\pi}{2} . \end{aligned}

If \(x=a(1-\cos \theta), y=a(\theta+\sin \theta)\), prove that \(\frac{d^{2} y}{d x^{2}}=-\frac{1}{a}\) at \(\theta=\frac{\pi}{2}\)

If \(y=x \log \left(\frac{x}{a+b x}\right)\), prove that \(x^{3} \frac{d^{2} y}{d x^{2}}=\left(x \frac{d y}{d x}-y\right)^{2}\).

If \(y=x \sin x\), then prove that, \(x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+\left(x^{2}+2\right) y=0\)

There is a polynomial \(P(x)=a x^{3}+b x^{2}+c x+d\) such that \(P(0)=P(1)=-2, P^{\prime}(0)=-1\), then find the value of \(a+b+c+d+10\)

\begin{aligned} &\text { If } f(x)=x+\tan x \text { and } g \text { is the inverse of } f \text {, then }\\\ &\text { prove that } g^{\prime}(x)=\frac{1}{2+\tan ^{2}(g(x))} . \end{aligned}

Let \(f(x)=1+x^{3}\). If \(g(x)=f^{-1}(x)\), then prove that \(g^{\prime \prime \prime}(2)=\frac{8}{3}\)

Suppose \(f\) is a differentiable function such that \(f(g(x))=x\) and \(f^{\prime}(x)=1+(f(x))^{2}\), then prove that \(g^{\prime}(x)=\frac{1}{1+x^{2}}\)