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

The derivative of ((3x^2-5^4)/((4x^3+2)^5)) is (3x(3x^2-5)^3(-7x^3+25x+4))/(16(2x^3+1)^6)

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

The first derivative of ((3x^2-5^4)/((4x^3+2)^5)) is (3x(3x^2-5)^3(-7x^3+25x+4))/(16(2x^3+1)^6)

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

\frac{3 x ( 3 x ^{2} - 5 ) ^{3} ( - 7 x ^{3} + 25 x + 4 )}{16 ( 2 x ^{3} + 1 ) ^{6}}

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} ((3 x^{2} - 5)^{4}) = 24 x (3 x^{2} - 5)^{3}

\frac{d}{d x} ((4 x^{3} + 2)^{5}) = 60 x^{2} (4 x^{3} + 2)^{4}

= \frac{24 x ( 3 x ^{2} - 5 ) ^{3} ( 4 x ^{3} + 2 ) ^{5} - 60 x ^{2} ( 4 x ^{3} + 2 ) ^{4} ( 3 x ^{2} - 5 ) ^{4}}{( ( 4 x ^{3} + 2 ) ^{5} ) ^{2}}

Simplify

\frac{24 x ( 3 x ^{2} - 5 ) ^{3} ( 4 x ^{3} + 2 ) ^{5} - 60 x ^{2} ( 4 x ^{3} + 2 ) ^{4} ( 3 x ^{2} - 5 ) ^{4}}{( ( 4 x ^{3} + 2 ) ^{5} ) ^{2}} : \frac{3 x ( 3 x ^{2} - 5 ) ^{3} ( - 7 x ^{3} + 25 x + 4 )}{16 ( 2 x ^{3} + 1 ) ^{6}}

= \frac{3 x ( 3 x ^{2} - 5 ) ^{3} ( - 7 x ^{3} + 25 x + 4 )}{16 ( 2 x ^{3} + 1 ) ^{6}}

y = - x^{2} + 4 x

y^{^{''}} + 2 y^{^{'}} + y = sin (x)

\int \frac{x ^{4} + 2 x ^{2} - 1}{x} d x

\frac{d}{d x} (\frac{sin ( x ) + cos ( x )}{x ^{3} + x - 2})

n = 21 \sum \infty e^{- 0.06 n}

What is the derivative of ((3x^2-5^4)/((4x^3+2)^5)) ?
The derivative of ((3x^2-5^4)/((4x^3+2)^5)) is (3x(3x^2-5)^3(-7x^3+25x+4))/(16(2x^3+1)^6)
What is the first derivative of ((3x^2-5^4)/((4x^3+2)^5)) ?
The first derivative of ((3x^2-5^4)/((4x^3+2)^5)) is (3x(3x^2-5)^3(-7x^3+25x+4))/(16(2x^3+1)^6)