What is the derivative of (x^5/((x^2+1)(x^3+4))) ?

The derivative of (x^5/((x^2+1)(x^3+4))) is (2x^7+12x^6+20x^4)/((x^2+1)^2(x^3+4)^2)

What is the first derivative of (x^5/((x^2+1)(x^3+4))) ?

The first derivative of (x^5/((x^2+1)(x^3+4))) is (2x^7+12x^6+20x^4)/((x^2+1)^2(x^3+4)^2)

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\frac{d}{d x} (\frac{x ^{5}}{( x ^{2} + 1 ) ( x ^{3} + 4 )})

\frac{2 x ^{7} + 12 x ^{6} + 20 x ^{4}}{( x ^{2} + 1 ) ^{2} ( x ^{3} + 4 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{x ^{5}}{( x ^{2} + 1 ) ( x ^{3} + 4 )})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( x ^{5} ) ( x ^{2} + 1 ) ( x ^{3} + 4 ) - \frac{d}{d x} ( ( x ^{2} + 1 ) ( x ^{3} + 4 ) ) x ^{5}}{( ( x ^{2} + 1 ) ( x ^{3} + 4 ) ) ^{2}}

\frac{d}{d x} (x^{5}) = 5 x^{4}

\frac{d}{d x} ((x^{2} + 1) (x^{3} + 4)) = 5 x^{4} + 3 x^{2} + 8 x

= \frac{5 x ^{4} ( x ^{2} + 1 ) ( x ^{3} + 4 ) - ( 5 x ^{4} + 3 x ^{2} + 8 x ) x ^{5}}{( ( x ^{2} + 1 ) ( x ^{3} + 4 ) ) ^{2}}

Simplify

\frac{5 x ^{4} ( x ^{2} + 1 ) ( x ^{3} + 4 ) - ( 5 x ^{4} + 3 x ^{2} + 8 x ) x ^{5}}{( ( x ^{2} + 1 ) ( x ^{3} + 4 ) ) ^{2}} : \frac{2 x ^{7} + 12 x ^{6} + 20 x ^{4}}{( x ^{2} + 1 ) ^{2} ( x ^{3} + 4 ) ^{2}}

= \frac{2 x ^{7} + 12 x ^{6} + 20 x ^{4}}{( x ^{2} + 1 ) ^{2} ( x ^{3} + 4 ) ^{2}}

\frac{d}{d x} (m sin (2 (m x + n y (x))))

x^{2} tan (x)

x \to + 0 lim (\frac{tan ( 6 x )}{sin ( 4 x )})

\int (x^{2} + 1) 5^{x^{3} + 3 x} d x

(4 + 3 x)^{^{'}}

What is the derivative of (x^5/((x^2+1)(x^3+4))) ?
The derivative of (x^5/((x^2+1)(x^3+4))) is (2x^7+12x^6+20x^4)/((x^2+1)^2(x^3+4)^2)
What is the first derivative of (x^5/((x^2+1)(x^3+4))) ?
The first derivative of (x^5/((x^2+1)(x^3+4))) is (2x^7+12x^6+20x^4)/((x^2+1)^2(x^3+4)^2)