Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Find the derivative using the quotient rule
- Find the derivative using the definition
- Find the derivative using the product rule
- Find the derivative using logarithmic differentiation
- Find the derivative
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
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To derive the function $x^{\ln\left(x\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation
Apply natural logarithm to both sides of the equality
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
Derive both sides of the equality with respect to $x$
When multiplying two powers that have the same base ($\ln\left(x\right)$), you can add the exponents
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
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 $\displaystyle f'(x)=\frac{a'}{a}$
The derivative of the linear function is equal to $1$
The derivative of the linear function is equal to $1$
Multiply the fraction and term
Multiply both sides of the equation by $y$
Substitute $y$ for the original function: $x^{\ln\left(x\right)}$
Simplify the fraction $\frac{2x^{\ln\left(x\right)}\ln\left(x\right)}{x}$ by $x$
The derivative of the function results in