What is the derivative of f(x)= 4/(sqrt((x^2+6x)^3)) ?

The derivative of f(x)= 4/(sqrt((x^2+6x)^3)) is -(6(x^2+6x)^2(2x+6))/(((x^2+6x)^3)^{3/2)}

What is the first derivative of f(x)= 4/(sqrt((x^2+6x)^3)) ?

The first derivative of f(x)= 4/(sqrt((x^2+6x)^3)) is -(6(x^2+6x)^2(2x+6))/(((x^2+6x)^3)^{3/2)}

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derivative of

f (x) = \frac{4}{( x ^{2} + 6 x ) ^{3}}

- \frac{6 ( x ^{2} + 6 x ) ^{2} ( 2 x + 6 )}{( ( x ^{2} + 6 x ) ^{3} ) ^{\frac{3}{2}}}

Solution steps

\frac{d}{d x} \frac{4}{( x ^{2} + 6 x ) ^{3}}

Take the constant out:

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

= 4 \frac{d}{d x} \frac{1}{( x ^{2} + 6 x ) ^{3}}

Apply exponent rule:

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

= 4 \frac{d}{d x} (((x^{2} + 6 x)^{3})^{- \frac{1}{2}})

Apply the chain rule:

- \frac{1}{2 ( ( x ^{2} + 6 x ) ^{3} ) ^{\frac{3}{2}}} \frac{d}{d x} ((x^{2} + 6 x)^{3})

= - \frac{1}{2 ( ( x ^{2} + 6 x ) ^{3} ) ^{\frac{3}{2}}} \frac{d}{d x} ((x^{2} + 6 x)^{3})

\frac{d}{d x} ((x^{2} + 6 x)^{3}) = 3 (x^{2} + 6 x)^{2} (2 x + 6)

= 4 - \frac{1}{2 ( ( x ^{2} + 6 x ) ^{3} ) ^{\frac{3}{2}}} \cdot 3 (x^{2} + 6 x)^{2} (2 x + 6)

Simplify

4 - \frac{1}{2 ( ( x ^{2} + 6 x ) ^{3} ) ^{\frac{3}{2}}} \cdot 3 (x^{2} + 6 x)^{2} (2 x + 6) : - \frac{6 ( x ^{2} + 6 x ) ^{2} ( 2 x + 6 )}{( ( x ^{2} + 6 x ) ^{3} ) ^{\frac{3}{2}}}

= - \frac{6 ( x ^{2} + 6 x ) ^{2} ( 2 x + 6 )}{( ( x ^{2} + 6 x ) ^{3} ) ^{\frac{3}{2}}}

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What is the derivative of f(x)= 4/(sqrt((x^2+6x)^3)) ?
The derivative of f(x)= 4/(sqrt((x^2+6x)^3)) is -(6(x^2+6x)^2(2x+6))/(((x^2+6x)^3)^{3/2)}
What is the first derivative of f(x)= 4/(sqrt((x^2+6x)^3)) ?
The first derivative of f(x)= 4/(sqrt((x^2+6x)^3)) is -(6(x^2+6x)^2(2x+6))/(((x^2+6x)^3)^{3/2)}