How do you differentiate ${ln x}^{\frac{1}{3}}$ $ln x^(1/3)$ ?

Question

How do you differentiate ${ln x}^{\frac{1}{3}}$ $ln x^(1/3)$ ?

1 Answer

Impact of this question

9514 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-07-29T13:39:30

$\frac{1}{3 x}$ $1/(3x)$

Explanation:

There are two methods: one that simplifies the function first, and the other which doesn't.

Without simplifying the function:

$y = ln (x^{\frac{1}{3}})$ $y=ln(x^(1/3))$

We will need to use the chain rule. Since:

$\frac{d}{d x} ln (x) = \frac{1}{x}$ $d/dxln(x)=1/x$

The chain rule tells us that:

$\frac{d}{d x} ln (u) = \frac{1}{u} \cdot \frac{d u}{d x}$ $d/dxln(u)=1/u*(du)/dx$

Thus:

$\frac{d y}{d x} = \frac{1}{x^{\frac{1}{3}}} \cdot \frac{d}{d x} x^{\frac{1}{3}}$ $dy/dx=1/x^(1/3)*d/dxx^(1/3)$

Use the power rule to find $\frac{d}{d x} x^{\frac{1}{3}}$ $d/dxx^(1/3)$ :

$\frac{d y}{d x} = \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{3} x^{- \frac{2}{3}}$ $dy/dx=1/x^(1/3)*1/3x^(-2/3)$

Simplify:

$\frac{d y}{d x} = \frac{1}{3} \cdot \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{2}{3}}}$ $dy/dx=1/3*1/x^(1/3)*1/x^(2/3)$

$\frac{d y}{d x} = \frac{1}{3} \cdot \frac{1}{x^{\frac{3}{3}}}$ $dy/dx=1/3*1/x^(3/3)$

$\frac{d y}{d x} = \frac{1}{3 x}$ $dy/dx=1/(3x)$

Alternatively, simplify first:

You may remember a rule of logarithms that states that:

$log (a^{b}) = b \cdot log (a)$ $log(a^b)=b*log(a)$

Thus:

$ln (x^{\frac{1}{3}}) = \frac{1}{3} \cdot ln (x)$ $ln(x^(1/3))=1/3*ln(x)$

So:

$y = \frac{1}{3} \cdot ln (x)$ $y=1/3*ln(x)$

Now when differentiating, we won't have to use the chain rule, and the $\frac{1}{3}$ $1/3$ is simply brought out of the differentiation:

$\frac{d y}{d x} = \frac{1}{3} \cdot \frac{d}{d x} ln (x)$ $dy/dx=1/3*d/dxln(x)$

$\frac{d y}{d x} = \frac{1}{3} \cdot \frac{1}{x}$ $dy/dx=1/3*1/x$

$\frac{d y}{d x} = \frac{1}{3 x}$ $dy/dx=1/(3x)$

Generalization:

This method can be applied to differentiating any function in the form:

$y = ln (x^{a})$ $y=ln(x^a)$

Since $y = a \cdot ln (x)$ $y=a*ln(x)$ , we know that $\frac{d y}{d x} = a \cdot \frac{1}{x} = \frac{a}{x}$ $dy/dx=a*1/x=a/x$ .

This can also be proven using the power rule, but above way is simpler.

Science

Math

Humanities

... and beyond

How do you differentiate ${ln x}^{\frac{1}{3}}$ $ln x^(1/3)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you differentiate lnx13ln x^(1/3)?

1 Answer

Explanation:

Related questions

Impact of this question

How do you differentiate ${ln x}^{\frac{1}{3}}$ $ln x^(1/3)$ ?