What is the derivative of (3sqrt(x))/(9x^2+4x-9) ?

The derivative of (3sqrt(x))/(9x^2+4x-9) is (3(-27x^2-4x-9))/(2sqrt(x)(9x^2+4x-9)^2)

What is the first derivative of (3sqrt(x))/(9x^2+4x-9) ?

The first derivative of (3sqrt(x))/(9x^2+4x-9) is (3(-27x^2-4x-9))/(2sqrt(x)(9x^2+4x-9)^2)

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derivative of

\frac{3 x}{9 x ^{2} + 4 x - 9}

\frac{3 ( - 27 x ^{2} - 4 x - 9 )}{2 x ( 9 x ^{2} + 4 x - 9 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{3 x}{9 x ^{2} + 4 x - 9})

Take the constant out:

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

= 3 \frac{d}{d x} (\frac{x}{9 x ^{2} + 4 x - 9})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 3 \cdot \frac{\frac{d}{d x} ( x ) ( 9 x ^{2} + 4 x - 9 ) - \frac{d}{d x} ( 9 x ^{2} + 4 x - 9 ) x}{( 9 x ^{2} + 4 x - 9 ) ^{2}}

\frac{d}{d x} (x) = \frac{1}{2 x}

\frac{d}{d x} (9 x^{2} + 4 x - 9) = 18 x + 4

= 3 \cdot \frac{\frac{1}{2 x} ( 9 x ^{2} + 4 x - 9 ) - ( 18 x + 4 ) x}{( 9 x ^{2} + 4 x - 9 ) ^{2}}

Simplify

3 \cdot \frac{\frac{1}{2 x} ( 9 x ^{2} + 4 x - 9 ) - ( 18 x + 4 ) x}{( 9 x ^{2} + 4 x - 9 ) ^{2}} : \frac{3 ( - 27 x ^{2} - 4 x - 9 )}{2 x ( 9 x ^{2} + 4 x - 9 ) ^{2}}

= \frac{3 ( - 27 x ^{2} - 4 x - 9 )}{2 x ( 9 x ^{2} + 4 x - 9 ) ^{2}}

x \to 1 lim (\frac{[ x ]}{∣ x ∣})

x \to π lim (sin (6 x))

x (3 x + 4)^{2}

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

What is the derivative of (3sqrt(x))/(9x^2+4x-9) ?
The derivative of (3sqrt(x))/(9x^2+4x-9) is (3(-27x^2-4x-9))/(2sqrt(x)(9x^2+4x-9)^2)
What is the first derivative of (3sqrt(x))/(9x^2+4x-9) ?
The first derivative of (3sqrt(x))/(9x^2+4x-9) is (3(-27x^2-4x-9))/(2sqrt(x)(9x^2+4x-9)^2)