Question

What is the derivative of x to the minus 5 power

Answer 1

$(x^{-5})'=-5x^{-6}$

Answer 2

Using implicit differentiation:

$y={ x }^{ -5 }\\ \\ y=\frac { 1 }{ { x }^{ 5 } } \\ \\ { x }^{ 5 }\cdot y=\frac { 1 }{ { x }^{ 5 } } \cdot { x }^{ 5 }$

$\\ \\ { x }^{ 5 }\cdot y=1\\ \\ { x }^{ 5 }\frac { dy }{ dx } +5{ x }^{ 4 }\cdot y=0$

$\\ \\ { x }^{ 5 }\frac { dy }{ dx } =-5{ x }^{ 4 }\cdot y\\ \\ { x }^{ 5 }\frac { dy }{ dx } =-5{ x }^{ 4 }\cdot \frac { 1 }{ { x }^{ 5 } }$

$\\ \\ \frac { 1 }{ { x }^{ 5 } } \cdot { x }^{ 5 }\frac { dy }{ dx } =-5{ x }^{ 4 }\cdot \frac { 1 }{ { x }^{ 5 } } \cdot \frac { 1 }{ { x }^{ 5 } }$

$\\ \\ \frac { dy }{ dx } =-\frac { 5{ x }^{ 4 } }{ { x }^{ 10 } } \\ \\ =-\frac { 5 }{ { x }^{ 6 } }$