If y = ({x^x})^x,then frac{dy}{dx} = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $y = ({x^x})^x$. Using the property of exponents $(a^m)^n = a^{mn}$,we have $y = x^{x^2}$. Taking the natural logarithm on both sides: $\ln

Vedclass · Accepted Answer

$x({x^x})^x(1 + 2\log x)$

Vedclass · Answer

(C) Given $y = ({x^x})^x$. Using the property of exponents $(a^m)^n = a^{mn}$,we have $y = x^{x^2}$. Taking the natural logarithm on both sides: $\ln y = \ln(x^{x^2}) = x^2 \ln x$. Differentiating both sides with respect to $x$ using the product rule: $\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x^2) \cdot \ln x + x^2 \cdot \frac{d}{dx}(\ln x)$. $\frac{1}{y} \frac{dy}{dx} = 2x \ln x + x^2 \cdot \frac{1}{x} = 2x \ln x + x$. $\frac{1}{y} \frac{dy}{dx} = x(1 + 2 \ln x)$. Therefore,$\frac{dy}{dx} = y \cdot x(1 + 2 \ln x) = x({x^x})^x(1 + 2 \ln x)$.

If $y = ({x^x})^x$,then $\frac{dy}{dx} =$

Similar Questions

If ${x^m}{y^n} = 2{(x + y)^{m + n}},$ the value of $\frac{dy}{dx}$ is

If $f(x)=\frac{(x+1) \sinh x}{e^{2 x} \tan x}$ and $\frac{f^{\prime}(x)}{f(x)}=\frac{1}{x+1}+\operatorname{coth} x+g(x)$, then $g(x)=$

Find $\frac{dy}{dx}$ for the function $(\cos x)^{y}=(\cos y)^{x}$.

If $\frac{d}{d x} \left[ \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x} \right] = f(x) \left[ \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+4} - 1 \right]$,then $f(5) = $

If $f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)$,then $f^{\prime}(1)=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module