Question

if y=x^x, then dy/dx is equal to​

Answer 1

Step-by-step explanation:

Given:

$y = {x}^{x}$

To find: dy/dx

Solution:

Whenever x is in power of x or any function of x,then power rule of Differentiation will not work.

To find solution of such problems,always take logarithm both sides and do differentiation in both sides of the equation.

$y = {x}^{x} \\ \\ \text{take \: log \: both \: sides} \\ \\ log \: y = log( {x}^{x} ) \\ \\ \text{apply \: log \: formula} \\ \\ \boxed{ log( {x}^{a} ) = a \: log \: x} \\ \\ log \: y = x \: log( {x})...eq1 \\ \\ \text{do \: differentiation \: of \: eq1} \\ \\ \frac{1}{y} \frac{dy}{dx} = x. \frac{1}{x} + log \: x.1 ...eq2\\ \\ \because \: \text{both \: are \: functions \: of \: x \: thus \: uv \: formula \: has \: applied} \\ \\ \boxed{ \frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx} }$

Simplify eq 2

$\frac{1}{y} . \frac{dy}{dx} = 1 + log \: x \\ \\ or \\ \\ \frac{dy}{dx} = y(1 + log \: x) \\ \\ or \\ \\ \boxed{ \bold{ \green{\frac{dy}{dx} = {x}^{x} (1 + log \: x)}}}$

is the final answer.

Always remember

${x}^{a} , {a}^{x} \: and \: {x}^{x}$

lies under different category of Differentiation.

Hope it helps you.

To learn more on brainly:

Differentiate w.r.t.x. : cos² (log (2x + 3))

https://brainly.in/question/6633326