1
Solved example of advanced differentiation
$\frac{d}{dx}\left(sin\left(x\right)^{ln\left(x\right)}\right)$
2
To derive the function $\sin\left(x\right)^{\ln\left(x\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation
$y=\sin\left(x\right)^{\ln\left(x\right)}$
3
Apply natural logarithm to both sides of the equality
$\ln\left(y\right)=\ln\left(\sin\left(x\right)^{\ln\left(x\right)}\right)$
4
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
$\ln\left(y\right)=\ln\left(x\right)\ln\left(\sin\left(x\right)\right)$
5
Derive both sides of the equality with respect to $x$
$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x\right)\ln\left(\sin\left(x\right)\right)\right)$
6
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(x\right)$ and $g=\ln\left(\sin\left(x\right)\right)$
$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x\right)\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)$
Intermediate steps
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 $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$
7
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 $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$
Explain this step further
Intermediate steps
The derivative of the linear function is equal to $1$
$1\left(\frac{1}{y}\right)$
Any expression multiplied by $1$ is equal to itself
$\frac{1}{y}$
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The derivative of the linear function is equal to $1$
$\frac{y^{\prime}}{y}=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$
Explain this step further
Intermediate steps
The derivative of the linear function is equal to $1$
$1\left(\frac{1}{y}\right)$
Any expression multiplied by $1$ is equal to itself
$\frac{1}{y}$
The derivative of the linear function is equal to $1$
$1\left(\frac{1}{x}\right)\ln\left(\sin\left(x\right)\right)$
Any expression multiplied by $1$ is equal to itself
$\frac{1}{x}\ln\left(\sin\left(x\right)\right)$
9
The derivative of the linear function is equal to $1$
$\frac{y^{\prime}}{y}=\frac{1}{x}\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$
Explain this step further
10
Multiply the fraction and term
$\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$
11
The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$
$\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{d}{dx}\left(x\right)\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$
Intermediate steps
The derivative of the linear function is equal to $1$
$1\left(\frac{1}{y}\right)$
Any expression multiplied by $1$ is equal to itself
$\frac{1}{y}$
The derivative of the linear function is equal to $1$
$1\left(\frac{1}{x}\right)\ln\left(\sin\left(x\right)\right)$
Any expression multiplied by $1$ is equal to itself
$\frac{1}{x}\ln\left(\sin\left(x\right)\right)$
The derivative of the linear function is equal to $1$
$1\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$
Any expression multiplied by $1$ is equal to itself
$\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$
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The derivative of the linear function is equal to $1$
$\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$
Explain this step further
13
Multiply the fraction and term
$\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}$
14
Multiply both sides of the equation by $y$
$y^{\prime}=y\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}\right)$
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Substitute $y$ for the original function: $\sin\left(x\right)^{\ln\left(x\right)}$
$y^{\prime}=\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}\right)\sin\left(x\right)^{\ln\left(x\right)}$
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The derivative of the function results in
$\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}\right)\sin\left(x\right)^{\ln\left(x\right)}$
Intermediate steps
Apply the trigonometric identity: $\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}$$=\cot\left(\theta \right)$
$\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\cot\left(x\right)\right)\sin\left(x\right)^{\ln\left(x\right)}$
17
Simplify the derivative
$\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\cot\left(x\right)\right)\sin\left(x\right)^{\ln\left(x\right)}$
Explain this step further
Final Answer
$\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\cot\left(x\right)\right)\sin\left(x\right)^{\ln\left(x\right)}$