What is the d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) ?

The d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) is -({y)/x}{({x}^2+{y}^2+{z}^2)^{3/2}}

What is the first d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) ?

The first d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) is -({y)/x}{({x}^2+{y}^2+{z}^2)^{3/2}}

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\frac{d}{d x} (\frac{y}{x ^{2} + y ^{2} + z ^{2}})

- \frac{y x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

Solution steps

\frac{d}{d x} (\frac{y}{x ^{2} + y ^{2} + z ^{2}})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= y \frac{d}{d x} (\frac{1}{x ^{2} + y ^{2} + z ^{2}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= y \frac{d}{d x} ((x^{2} + y^{2} + z^{2})^{- \frac{1}{2}})

Apply the chain rule:

- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

= - \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

\frac{d}{d x} (x^{2} + y^{2} + z^{2}) = 2 x

= y (- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 x)

Simplify

y (- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 x) : - \frac{y x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

= - \frac{y x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

\frac{d y}{d x} = \frac{2 y}{3 x}

f (x) = 5 - 3 x, at x = - 2

x \to - 5 lim (4 x^{3} + x)

\frac{d y}{d x} + 3 y = 2

x \to 3 lim (- 8)

What is the d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) ?
The d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) is -({y)/x}{({x}^2+{y}^2+{z}^2)^{3/2}}
What is the first d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) ?
The first d/(d{x)}(({y})/(sqrt({x)^2+{y)^2+{z}^2}}) is -({y)/x}{({x}^2+{y}^2+{z}^2)^{3/2}}