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

The derivative of (3x^3+x+2/(5x^2+1)) is (15x^4+4x^2-20x+1)/((5x^2+1)^2)

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

The first derivative of (3x^3+x+2/(5x^2+1)) is (15x^4+4x^2-20x+1)/((5x^2+1)^2)

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

\frac{15 x ^{4} + 4 x ^{2} - 20 x + 1}{( 5 x ^{2} + 1 ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (3 x^{3} + x + 2) = 9 x^{2} + 1

\frac{d}{d x} (5 x^{2} + 1) = 10 x

= \frac{( 9 x ^{2} + 1 ) ( 5 x ^{2} + 1 ) - 10 x ( 3 x ^{3} + x + 2 )}{( 5 x ^{2} + 1 ) ^{2}}

Expand

(9 x^{2} + 1) (5 x^{2} + 1) - 10 x (3 x^{3} + x + 2) : 15 x^{4} + 4 x^{2} - 20 x + 1

= \frac{15 x ^{4} + 4 x ^{2} - 20 x + 1}{( 5 x ^{2} + 1 ) ^{2}}

What is the derivative of (3x^3+x+2/(5x^2+1)) ?
The derivative of (3x^3+x+2/(5x^2+1)) is (15x^4+4x^2-20x+1)/((5x^2+1)^2)
What is the first derivative of (3x^3+x+2/(5x^2+1)) ?
The first derivative of (3x^3+x+2/(5x^2+1)) is (15x^4+4x^2-20x+1)/((5x^2+1)^2)