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

The d/(d{y)}(({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}) is -(3{x)/y}{({x}^2+{y}^2+{z}^2)^{5/2}}

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

The first d/(d{y)}(({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}) is -(3{x)/y}{({x}^2+{y}^2+{z}^2)^{5/2}}

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

- \frac{3 x y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

Solution steps

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

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d y} (x^{2} + y^{2} + z^{2})

= - \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d y} (x^{2} + y^{2} + z^{2})

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

= x (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 y)

Simplify

x (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 y) : - \frac{3 x y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

= - \frac{3 x y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

\int \frac{1}{x x ^{2} - π} d x

\int_{0}^{1} e^{x} - x d x

\int x sin (3 x) d x

y = 2 cos (x), y = 6 - 4 cos (x), 0 \leq x \leq 2 π

f (x) = 18 x^{2} - 14 x + 9

What is the d/(d{y)}(({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}) ?
The d/(d{y)}(({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}) is -(3{x)/y}{({x}^2+{y}^2+{z}^2)^{5/2}}
What is the first d/(d{y)}(({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}) ?
The first d/(d{y)}(({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}) is -(3{x)/y}{({x}^2+{y}^2+{z}^2)^{5/2}}