If y = x^{ln x},then dy/dx equals :- | vedclass

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

Question

(A) Given $y = x^{\ln x}$.Taking natural logarithm on both sides,we get $\ln y = \ln(x^{\ln x})$.Using the property $\ln(a^b) = b \ln a$,we have $\ln

Vedclass · Accepted Answer

$2 \ln x \cdot x^{\ln x - 1}$

Vedclass · Answer

(A) Given $y = x^{\ln x}$.Taking natural logarithm on both sides,we get $\ln y = \ln(x^{\ln x})$.Using the property $\ln(a^b) = b \ln a$,we have $\ln y = \ln x \cdot \ln x = (\ln x)^2$.Differentiating both sides with respect to $x$:$\frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}[(\ln x)^2]$.Using the chain rule,$\frac{d}{dx}[(\ln x)^2] = 2 \ln x \cdot \frac{d}{dx}(\ln x) = 2 \ln x \cdot \frac{1}{x}$.Therefore,$\frac{1}{y} \cdot \frac{dy}{dx} = \frac{2 \ln x}{x}$.Multiplying by $y$,we get $\frac{dy}{dx} = y \cdot \frac{2 \ln x}{x}$.Substituting $y = x^{\ln x}$,we get $\frac{dy}{dx} = x^{\ln x} \cdot \frac{2 \ln x}{x} = 2 \ln x \cdot x^{\ln x - 1}$.

If $y = x^{\ln x}$,then $dy/dx$ equals :-

Similar Questions

If $h(x) = x^{x^x}$,then at $x = 1$,$\frac{h^{\prime}(x)}{h(x)}$ is equal to

If $f(\theta) = \cos \theta_1 \cdot \cos \theta_2 \cdot \cos \theta_3 \cdots \cos \theta_n$,then $\tan \theta_1 + \tan \theta_2 + \tan \theta_3 + \cdots + \tan \theta_n =$

If $y = (1 + x)^x$,then $\frac{dy}{dx} = $

Differentiate the function with respect to $x$: $(5x)^{3 \cos 2x}$

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module