Question

Let $y=t^{10}+1$ and $x=t^8+1,$ then $\frac{d^{2}y}{d{x}^2}$ is

• A. $\frac{5t}{2}$
• B. $20t^8$
• C. $\frac{5}{16}{t}^6$
• D. None of these.

Answer 1

The correct option is A

$\frac{5t}{2}$

Explanation for the correct option:

Differentiating the given equations:

Given

$y=t^{10}+1$

Differentiate both w.r.t $t$

$\frac{dy}{dt}=10t^9\dots \dots \left(1\right)\left[\because \frac{d}{dx}{x}^n={\mathrm{nx}}^{n-1}\right]$

Also, $x=t^8+1,$

Differentiate both w.r.t $t$

$\frac{dx}{dt}=8t^7\dots \dots \left(2\right)$

By dividing equations $\left(1\right)\text{and}\left(2\right)$ we get

$\begin{array}{rcl}\frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}& =& \frac{10t^9}{8t^7}\\ & \Rightarrow & \frac{dy}{dx}=\frac{5}{4}{t}^2\\ & \Rightarrow & \frac{d^{2}y}{d{x}^2}=\frac{5t}{2}\end{array}$

Therefore, $\begin{array}{rcl}\frac{d^{2}y}{d{x}^2}& =& \frac{5t}{2}\end{array}$

Hence, the correct answer is option (A).