Embibe Experts Solutions for Chapter: Differential Coefficient, Exercise 1: JEE Main - 25 June 2022 Shift 1

For the curve $C:\left(x^2+y^2-3\right)+{\left(x^2-y^2-1\right)}^5=0$, the value of $3y^{\text{'}}-y^3{y}^{\text{'}\text{'}}$, at the point $\left(\alpha ,\alpha \right),\alpha >0$, on $C$, is equal to ________.

Let $x\left(t\right)=2\sqrt{2}\cos t\sqrt{\sin 2t}$ and $y\left(t\right)=2\sqrt{2}\sin t\sqrt{\sin 2t},t\in \left(0,\frac{\pi }{2}\right)$. Then $\frac{1+{\left(\frac{dy}{dx}\right)}^2}{\frac{d^{2}y}{d{x}^2}}$ at $t=\frac{\pi }{4}$ is equal to

If $f\left(x\right)=x^3-x^2{f}^{\text{'}}\left(1\right)+x{f}^{"}\left(2\right)-f^{\text{'}\text{'}\text{'}}\left(3\right)$, $x\in \mathrm{R}$, then

Let $y\left(x\right)=\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)$. Then $y^{\text{'}}-y^{"}$ at $x=-1$ is equal to

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function that satisfies the relation $f(x+y)=f\left(x\right)+f\left(y\right)-1,\forall x$, $y\in \mathrm{ℝ}$. If $f^{\text{'}}\left(0\right)=2$, then $\left|f\right(-2\left)\right|$ is equal to

Let $y=f\left(x\right)={\sin }^3\left(\frac{\pi }{3}\left(\cos \left(\frac{\pi }{3\sqrt{2}}{\left(-4x^3+5x^2+1\right)}^{\frac{3}{2}}\right)\right)\right)$. Then, at $x=1$,

If $f\left(x\right)=x^2+g^{\text{'}}\left(1\right)x+g^{"}\left(2\right)$ and $g\left(x\right)=f\left(1\right)x^2+x{f}^{\text{'}}\left(x\right)+f^{"}\left(x\right)$, then the value of $f\left(4\right)-g\left(4\right)$ is equal to _____ .

If $y\left(x\right)=x^x,x>0$, then $y^{"}\left(2\right)-2y^{\text{'}}\left(2\right)$ is equal to :