What is the derivative of ((cos(x)-cos^3(x)))/(x^2) ?

The derivative of ((cos(x)-cos^3(x)))/(x^2) is (sin(x)(x(-3sin^2(x)+2)-sin(2x)))/(x^3)

What is the first derivative of ((cos(x)-cos^3(x)))/(x^2) ?

The first derivative of ((cos(x)-cos^3(x)))/(x^2) is (sin(x)(x(-3sin^2(x)+2)-sin(2x)))/(x^3)

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derivative of

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{6}{2 - 4 x}

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

\frac{50}{( x + 5 )}

\frac{( ( x ) sin ( x ) )}{( x + 1 )}

y = (2 x - 1) a^{x}

What is the derivative of ((cos(x)-cos^3(x)))/(x^2) ?
The derivative of ((cos(x)-cos^3(x)))/(x^2) is (sin(x)(x(-3sin^2(x)+2)-sin(2x)))/(x^3)
What is the first derivative of ((cos(x)-cos^3(x)))/(x^2) ?
The first derivative of ((cos(x)-cos^3(x)))/(x^2) is (sin(x)(x(-3sin^2(x)+2)-sin(2x)))/(x^3)