An online logarithmic differentiation calculator to differentiate a function by taking a log derivative.
Free Logarithms Calculator - Simplify logarithmic expressions using algebraic rules step-by-step. This website uses cookies to ensure you get the best experience. Derivatives Derivative Applications Limits Integrals Integral Applications Riemann Sum Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. The Derivative Calculator supports solving first, second., fourth derivatives, as well as implicit differentiation and finding the zeros/roots. You can also get a better visual and understanding of the function by using our graphing tool. Chain Rule: d d x f (g (x)) = f ' (g (x)) g ' (x) Step 2: Click the blue arrow to submit. The following variables and constants are reserved: e = Euler's number, the base of the exponential function (2.718281.); i = imaginary number (i ² = -1); pi, π = the ratio of a circle's circumference to its diameter (3.14159.); phi, Φ = the golden ratio (1,6180.); You can enter expressions the same way you see them in your math textbook. Implicit multiplication (5x = 5.x) is supported.
Log Derivative Calculator
An online logarithmic differentiation calculator to differentiate a function by taking a log derivative.
Use our free Logarithmic differentiation calculator to find the differentiation of the given function based on the logarithms. Logarithmic differentiation is a method used to differentiate functions by employing the logarithmic derivative of a function. It is particularly useful for functions where a variable is raised to a variable power and to differentiate the logarithm of a function rather than the function itself. In the logarithmic differentiation calculator enter a function to differentiate.
Logarithmic differentiation is differentiating algebraically complicated functions or functions for which the ordinary rules of differentiation do not apply.
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The method of differentiating functions by first taking logarithms and then differentiating is called logarithmic differentiation. We use logarithmic differentiation in situations where it is easier to differentiate the logarithm of a function than to differentiate the function itself. This approach allows calculating derivatives of power, rational and some irrational functions in an efficient manner.
Consider this method in more detail. Let (y = fleft( x right)). Take natural logarithms of both sides:
[ln y = ln fleft( x right).]
Next, we differentiate this expression using the chain rule and keeping in mind that (y) is a function of (x.)
[
{{left( {ln y} right)^prime } = {left( {ln fleft( x right)} right)^prime },;;}Rightarrow
{frac{1}{y}y’left( x right) = {left( {ln fleft( x right)} right)^prime }.}
]
It’s seen that the derivative is
[
{y’ = y{left( {ln fleft( x right)} right)^prime } }
= {fleft( x right){left( {ln fleft( x right)} right)^prime }.}
]
The derivative of the logarithmic function is called the logarithmic derivative of the initial function (y = fleft( x right).)
This differentiation method allows to effectively compute derivatives of power-exponential functions, that is functions of the form
[y = u{left( x right)^{vleft( x right)}},]
where (uleft( x right)) and (vleft( x right)) are differentiable functions of (x.)
In the examples below, find the derivative of the function (yleft( x right)) using logarithmic differentiation.
Solved Problems
Click or tap a problem to see the solution.
Example 2
[y = {x^{{frac{1}{x}}}},;x gt 0.]Example 4
[y = {x^{cos x}},;x gt 0.]Example 6
[{y = {x^{2x}};;}kern0pt{left( {x gt 0,;x ne 1} right)}]Example 7
[y = {left( {x – 1} right)^2}{left( {x – 3} right)^5}]Example 8
[y = {left( {x + 1} right)^2}{left( {x – 2} right)^4}]Example 9
[{y = {frac{{{{left( {x – 2} right)}^2}}}{{{{left( {x + 5} right)}^3}}}},;}kern0pt{x gt 2.}]Example 10
[{yleft( x right) }={ frac{{{{left( {x + 1} right)}^2}}}{{{{left( {x + 2} right)}^3}{{left( {x + 3} right)}^4}}},;;}kern-0.3pt{x gt – 1.}]Example 11
[y = sqrt[large xnormalsize]{x},;x gt 0.]Example 12
[y = {left( {ln x} right)^x},;x gt 1.]Use Logarithmic Differentiation Calculator
Example 13
[y = {left( {{e^x}} right)^{{e^x}}}.]Example 14
[y = {left( {ln x} right)^{ln x}},;x gt 1.]Example 15
[{y = {x^{{x^2}}};;}kern-0.3pt{left( {x gt 0,;x ne 1} right)}]Example 16
[{y = {x^{{x^n}}};;}kern-0.3pt{left( {x > 0,;x ne 1} right)}]Example 17
[{y = {x^{{2^x}}};;}kern-0.3pt{left( {x > 0,;x ne 1} right)}]Example 18
[{y = {2^{{x^x}}};;}kern-0.3pt{left( {x > 0,;x ne 1} right)}]Example 19
[{y = {x^{sqrt x }};;}kern-0.3pt{left( {x gt 0,;x ne 1} right)}]Example 20
[{y = {x^{{x^x}}};;}kern-0.3pt{left( {x gt 0,;x ne 1} right)}]Example 21
[{y = {sqrt x ^{sqrt x }};;}kern-0.3pt{left( {x gt 0,;x ne 1} right)}]Logarithmic Differentiation Calculator Calculus
Example 22
[y = {left( {sin x} right)^{cos x}}]Example 23
[{y = sqrt[3]{{{frac{{x – 2}}{{x + 2}}}}},;}kern0pt{x gt 2.}]Example 24
[{y = sqrt[large 3normalsize]{{frac{{{x^2} – 3}}{{1 + {x^5}}}}},;;}kern-0.3pt{x gt sqrt 3 .}]Example 25
[y = {left( {cos x} right)^{arcsin x}}]Example 26
[y = {left( {sin x} right)^{arctan x}}]Solution.
First we take logarithms of the left and right side of the equation:
[
{ln y = ln {x^x},;;}Rightarrow
{ln y = xln x.}
]
Now we differentiate both sides meaning that (y) is a function of (x:)
[
{{left( {ln y} right)^prime } = {left( {xln x} right)^prime },;;}Rightarrow
{frac{1}{y} cdot y’ = x’ln x + x{left( {ln x} right)^prime },;;}Rightarrow
{frac{{y’}}{y} = 1 cdot ln x + x cdot frac{1}{x},;;}Rightarrow
{frac{{y’}}{y} = ln x + 1,;;}Rightarrow
{y’ = yleft( {ln x + 1} right),;;}Rightarrow
{y’ = {x^x}left( {ln x + 1} right),;;}kern0pt{text{where};;x gt 0.}
]
Example 2.
[y = {x^{{frac{1}{x}}}},;x gt 0.]
Solution.
First we take logarithms of both sides:
[{ln y = ln {x^{frac{1}{x}}},};; Rightarrow {ln y = frac{1}{x}ln x.}]
Differentiate the last equation with respect to (x:)
[{left( {ln y} right)^prime = left( {frac{1}{x}ln x} right)^prime,};; Rightarrow {frac{1}{y} cdot y^prime = left( {frac{1}{x}} right)^primeln x + frac{1}{x}left( {ln x} right)^prime,};; Rightarrow {frac{{y^prime}}{y} = – frac{1}{{{x^2}}} cdot ln x + frac{1}{x} cdot frac{1}{x},};; Rightarrow {frac{{y^prime}}{y} = frac{1}{{{x^2}}}left( {1 – ln x} right),};; Rightarrow {y^prime = frac{y}{{{x^2}}}left( {1 – ln x} right).}]
Substitute the original function instead of (y) in the right-hand side:
[{y^prime = frac{{{x^{frac{1}{x}}}}}{{{x^2}}}left( {1 – ln x} right) }={ {x^{frac{1}{x} – 2}}left( {1 – ln x} right) }={ {x^{frac{{1 – 2x}}{x}}}left( {1 – ln x} right).}]
Solution.
Apply logarithmic differentiation:
[
{ln y = ln left( {{x^{ln x}}} right),;;}Rightarrow
{ln y = ln xln x = {ln ^2}x,;;}Rightarrow
{{left( {ln y} right)^prime } = {left( {{{ln }^2}x} right)^prime },;;}Rightarrow
{frac{{y’}}{y} = 2ln x{left( {ln x} right)^prime },;;}Rightarrow
{frac{{y’}}{y} = frac{{2ln x}}{x},;;}Rightarrow
{y’ = frac{{2yln x}}{x},;;}Rightarrow
{y’ = frac{{2{x^{ln x}}ln x}}{x} }={ 2{x^{ln x – 1}}ln x.}
]
Example 4.
[y = {x^{cos x}},;x gt 0.]Solution.
Take the logarithm of the given function:
[
{ln y = ln left( {{x^{cos x}}} right),;;}Rightarrow
{ln y = cos xln x.}
]
Differentiating the last equation with respect to (x,) we obtain:
[
{{left( {ln y} right)^prime } = {left( {cos xln x} right)^prime },;;}Rightarrow
{frac{1}{y} cdot y’ }={ {left( {cos x} right)^prime }ln x + cos x{left( {ln x} right)^prime },;;}Rightarrow
{{frac{{y’}}{y} }={ left( { – sin x} right) cdot ln x + cos x cdot frac{1}{x},;;}}Rightarrow
{{frac{{y’}}{y} }={ – sin xln x + frac{{cos x}}{x},;;}}Rightarrow
{{y’ }={ yleft( {frac{{cos x}}{x} – sin xln x} right).}}
]
Substitute the original function instead of (y) in the right-hand side:
[{y’ = {x^{cos x}}cdot}kern0pt{left( {frac{{cos x}}{x} – sin xln x} right),}]
where (x gt 0.)
Solution.
Taking logarithms of both sides, we get
[{ln y = ln {x^{arctan x}},};; Rightarrow {ln y = arctan xln x.}]
Differentiate this equation with respect to (x:)
[{left( {ln y} right)^prime = left( {arctan xln x} right)^prime,};; Rightarrow {frac{1}{y} cdot y^prime = left( {arctan x} right)^primeln x }+{ arctan xleft( {ln x} right)^prime,};; Rightarrow {frac{{y^prime}}{y} = frac{1}{{1 + {x^2}}} cdot ln x }+{ arctan x cdot frac{1}{x},};; Rightarrow {frac{{y^prime}}{y} = frac{{ln x}}{{1 + {x^2}}} }+{ frac{{arctan x}}{x},};; Rightarrow {y^prime = yleft( {frac{{ln x}}{{1 + {x^2}}} + frac{{arctan x}}{x}} right),}]
where (y = {x^{arctan x}}.)
Example 6.
[{y = {x^{2x}};;}kern0pt{left( {x gt 0,;x ne 1} right)}]Logarithmic Differentiation Calculator Emathhelp
Solution.
Taking logarithms of both sides, we can write the following equation:
[{ln y = ln {x^{2x}},;;} Rightarrow {ln y = 2xln x.}]
Further we differentiate the left and right sides:
[
{{left( {ln y} right)^prime } = {left( {2xln x} right)^prime },;;}Rightarrow
{frac{1}{y} cdot y’ }={ {left( {2x} right)^prime } cdot ln x + 2x cdot {left( {ln x} right)^prime },;;}Rightarrow
{frac{{y’}}{y} = 2 cdot ln x + 2x cdot frac{1}{x},;;}Rightarrow
{frac{{y’}}{y} = 2ln x + 2,;;}Rightarrow
{y’ = 2yleft( {ln x + 1} right);;}kern0pt{text{or};;y’ = 2{x^{2x}}left( {ln x + 1} right).}
]