What is the derivative of $-(5-x^2)^(1/2)$ ?

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Answer 1 · 2017-11-05T09:35:55

$x / sqrt(5-x^2)$

$d/dx (-sqrt(5-x^2))$ Refer to this as Expression [ A ]

We will pull the negative sign ( - ) out, and rewrite as

$- d/dx (sqrt(5-x^2))$ Refer to this as Expression [ B ]

We will use the Chain Rule now to move on, as there are two functions to deal with

Chain Rule states that $dy/dx = (dy/(du)) * ((du)/dx)$

In Expression [ A ], let $u = 5 - x^2$

We will rewrite the Expression [ B ] as

$- d/ (du) (sqrt(u)) * d/(dx) (5 - x ^ 2)$ Refer to this as Expression [ C ]

We can write $d /(du) (sqrt(u))$ as $1/(2*sqrt(u))$

Similarly, we can write $d /(dx) (5 - x^2)$ as $( -2*x)$

Using these two results in Expression [ C ], we obtain

$-1/(2 * sqrt(u)) * (-2*x)$

Substitute back $u = (5 - x^2)$

$-(1/(2 sqrt(5-x^2))) * (-2*x)$

This expression simplifies to

$(2*x)/(2*sqrt(5-x^2)$

This expression simplifies to

$x/(sqrt(5 - x^2)$ is our final answer.

What is the derivative of -(5-x^2)^(1/2) -(5-x^2)^(1/2) ?