Derivatives

Logarithmic Differentiation

Logarithmic differentiation allows us to take derivatives of more complicated expressions. The derivative rules give us a way to quickly differentiate expressions like $x^5$ , $e^{4x}$ , and $2^x$ But what about $x^x$ ?

Use a table of values to sketch $\displaystyle f(x)=x^x$ Include the listed negative inputs as individual real-valued points rather than joining them into a continuous curve

The expression $x^x$ at $x=0$ leads to the indeterminate form $0^0$ in a limiting context. We leave the function undefined at zero, even though its right-hand limit is 1. Because the real domain is complicated when $x<0$ , this function is usually considered only for $x>0$ in differentiation.

So, how do we determine its derivative? With logarithmic differentiation, we take the natural logarithm of both sides. Then we use log rules to simplify before differentiating implicitly.

\displaystyle y=x^x,\quad x>0

\displaystyle T(t)=1+(1)(t-(1))

Positive input x1

\displaystyle \ln y=x\ln x,\qquad y'=x^x(\ln x+1)

\displaystyle f(1)=1,\quad f'(1)=1

The slope changes sign at the minimum, where x = 1/e.

The source pairs the question $\displaystyle y=(\cos(\pi x))^{2x}$ with working for $\displaystyle y=x^x$ Differentiate both expressions, and distinguish the two results

If $\displaystyle y = \left(\cos(\pi x)\right)^{2x}$ , find $\displaystyle \dfrac{dy}{dx}$

Products, Quotients, and Absolute Values

It would be tedious to differentiate an expression like

y=\frac{(x-2)^2(3x+1)^3\sqrt{x^2+2}}{(x-1)(x+10)^{10}}

using repeated product and quotient rules, with a high risk of making mistakes. We can use logarithms to simplify the problem. Taking the absolute value first lets us use $\ln|y|$ when $y$ is negative as well as positive, provided $y\ne0$

QUESTION

For

\displaystyle x>0

, what is the derivative of

\displaystyle x^x

If $\displaystyle y = {\dfrac{(x-2)^2(3x+1)^3\sqrt{x^2+2}}{(x-1)(x+10)^{10}}}$ , find $\displaystyle \dfrac{dy}{dx}$

◀ Previous SectionRelated Rates Next Section ▶Differential Approximation

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Derivatives

Logarithmic Differentiation

Use a table of values to sketch $\displaystyle f(x)=x^x$ Include the listed negative inputs as individual real-valued points rather than joining them into a continuous curve

So, how do we determine its derivative? With logarithmic differentiation, we take the natural logarithm of both sides. Then we use log rules to simplify before differentiating implicitly.

\displaystyle y=x^x,\quad x>0

\displaystyle T(t)=1+(1)(t-(1))

Positive input x1

\displaystyle \ln y=x\ln x,\qquad y'=x^x(\ln x+1)

\displaystyle f(1)=1,\quad f'(1)=1

The slope changes sign at the minimum, where x = 1/e.

The source pairs the question $\displaystyle y=(\cos(\pi x))^{2x}$ with working for $\displaystyle y=x^x$ Differentiate both expressions, and distinguish the two results

If $\displaystyle y = \left(\cos(\pi x)\right)^{2x}$ , find $\displaystyle \dfrac{dy}{dx}$

Products, Quotients, and Absolute Values

It would be tedious to differentiate an expression like

y=\frac{(x-2)^2(3x+1)^3\sqrt{x^2+2}}{(x-1)(x+10)^{10}}

QUESTION

For

\displaystyle x>0

, what is the derivative of

\displaystyle x^x

If $\displaystyle y = {\dfrac{(x-2)^2(3x+1)^3\sqrt{x^2+2}}{(x-1)(x+10)^{10}}}$ , find $\displaystyle \dfrac{dy}{dx}$

◀ Previous SectionRelated Rates Next Section ▶Differential Approximation

Logarithmic Differentiation