Derivative of Natural Logarithm Function/Examples/(x-2)^1/3 times (x-3)^1/2 times (2x-1)^3/2

Examples of Use of Derivative of Natural Logarithm Function

Let $y = \paren {x - 2}^{1/3} \paren {x - 3}^{1/2} \paren {2 x - 1}^{3/2}$.

Then:

$\dfrac {\d y} {\d x} = \paren {\dfrac 1 {3 \paren {x - 2} } + \dfrac 1 {2 \paren {x - 3} } + \dfrac 3 {2 x - 1} } \paren {x - 2}^{1/3} \paren {x - 3}^{1/2} \paren {2 x - 1}^{3/2}$

Proof

\(\ds y\)

\(=\)

\(\ds \paren {x - 2}^{1/3} \paren {x - 3}^{1/2} \paren {2 x - 1}^{3/2}\)

\(\ds \leadsto \ \ \)

\(\ds \ln y\)

\(=\)

\(\ds \dfrac 1 3 \map \ln {x - 2} + \dfrac 1 2 \map \ln {x - 3} + \dfrac 3 2 \map \ln {2 x - 1}\)

Logarithm of Power, Sum of Logarithms

\(\ds \leadsto \ \ \)

\(\ds \dfrac 1 y \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \dfrac 1 {3 \paren {x - 2} } + \dfrac 1 {2 \paren {x - 3} } + \dfrac 3 2 \dfrac 2 {2 x - 1}\)

Derivative of Composite Function, Derivative of Natural Logarithm

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \paren {\dfrac 1 {3 \paren {x - 2} } + \dfrac 1 {2 \paren {x - 3} } + \dfrac 3 {2 x - 1} } \paren {x - 2}^{1/3} \paren {x - 3}^{1/2} \paren {2 x - 1}^{3/2}\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Variable Index: Example