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# Precalculus Calculator

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of logarithmic differentiation. This solution was automatically generated by our smart calculator:

$\frac{d}{dx}\left(x^x\right)$
2

To derive the function $x^x$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=x^x$
3

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(x^x\right)$

Apply logarithm properties to both sides of the equality

$\ln\left(y\right)=\ln\left(x^x\right)$

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\ln\left(y\right)=x\ln\left(x\right)$
4

Apply logarithm properties to both sides of the equality

$\ln\left(y\right)=x\ln\left(x\right)$
5

Derive both sides of the equality with respect to $x$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x\ln\left(x\right)\right)$
6

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x\right)\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$The derivative of the linear function is equal to$11\ln\left(x\right)$Any expression multiplied by$1$is equal to itself$\ln\left(x\right)$7 The derivative of the linear function is equal to$1\frac{d}{dx}\left(\ln\left(y\right)\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If$f(x)=ln\:a$(where$a$is a function of$x$), then$\displaystyle f'(x)=\frac{a'}{a}\frac{1}{y}\frac{d}{dx}\left(y\right)=\ln\left(x\right)+x\frac{1}{x}\frac{d}{dx}\left(x\right)$8 The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If$f(x)=ln\:a$(where$a$is a function of$x$), then$\displaystyle f'(x)=\frac{a'}{a}\frac{1}{y}\frac{d}{dx}\left(y\right)=\ln\left(x\right)+x\frac{1}{x}\frac{d}{dx}\left(x\right)$The derivative of the linear function is equal to$11\ln\left(x\right)$Any expression multiplied by$1$is equal to itself$\ln\left(x\right)$The derivative of the linear function is equal to$11\left(\frac{1}{y}\right)$Any expression multiplied by$1$is equal to itself$\frac{1}{y}$9 The derivative of the linear function is equal to$1\frac{y^{\prime}}{y}=\ln\left(x\right)+x\frac{1}{x}\frac{d}{dx}\left(x\right)$The derivative of the linear function is equal to$11\ln\left(x\right)$Any expression multiplied by$1$is equal to itself$\ln\left(x\right)$The derivative of the linear function is equal to$11\left(\frac{1}{y}\right)$Any expression multiplied by$1$is equal to itself$\frac{1}{y}$The derivative of the linear function is equal to$11x\frac{1}{x}$Any expression multiplied by$1$is equal to itself$x\frac{1}{x}$10 The derivative of the linear function is equal to$1\frac{y^{\prime}}{y}=\ln\left(x\right)+x\frac{1}{x}$11 Multiply the fraction and term$\frac{y^{\prime}}{y}=\ln\left(x\right)+\frac{x}{x}$12 Simplify the fraction$\frac{y^{\prime}}{y}=\ln\left(x\right)+1$13 Multiply both sides of the equation by$yy^{\prime}=\left(\ln\left(x\right)+1\right)y$14 Substitute$y$for the original function:$x^xy^{\prime}=\left(\ln\left(x\right)+1\right)x^x$15 The derivative of the function results in$\left(\ln\left(x\right)+1\right)x^x$##  Final answer to the problem$\left(\ln\left(x\right)+1\right)x^x\$

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