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Difficult Problems

1

Example

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\sqrt{x}$ and $g=\ln\left(x\right)$

$\sqrt{x}\cdot\frac{d}{dx}\left(\ln\left(x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(\sqrt{x}\right)$
3

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\sqrt{x}\cdot\frac{d}{dx}\left(\ln\left(x\right)\right)+\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)$
4

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\sqrt{x}\cdot\frac{1}{x}\cdot\frac{d}{dx}\left(x\right)+\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)$
5

The derivative of the linear function is equal to $1$

$\sqrt{x}\cdot\frac{1}{x}$
6

Multiplying the fraction and term

$\frac{\sqrt{x}}{x}+\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)$
7

Simplifying the fraction by $x$

$x^{-\frac{1}{2}}+\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)$
8

Using the power rule of logarithms

$x^{-\frac{1}{2}}+\ln\left(x^{\frac{1}{2}x^{-\frac{1}{2}}}\right)$
9

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

$x^{-\frac{1}{2}}+\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)$

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