What is the derivative of 1/2 (x-1^3sin(1/(x-1))) ?

The derivative of 1/2 (x-1^3sin(1/(x-1))) is 1/2 (3sin(1/(x-1))(x-1)^2-cos(1/(x-1))(x-1))

What is the first derivative of 1/2 (x-1^3sin(1/(x-1))) ?

The first derivative of 1/2 (x-1^3sin(1/(x-1))) is 1/2 (3sin(1/(x-1))(x-1)^2-cos(1/(x-1))(x-1))

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\frac{d}{d x} (\frac{1}{2} (x - 1)^{3} sin (\frac{1}{x - 1}))

\frac{1}{2} (3 sin (\frac{1}{x - 1}) (x - 1)^{2} - cos (\frac{1}{x - 1}) (x - 1))

Solution steps

\frac{d}{d x} (\frac{1}{2} (x - 1)^{3} sin (\frac{1}{x - 1}))

Take the constant out:

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

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (x - 1)^{3}, g = sin (\frac{1}{x - 1})

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

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

\frac{d}{d x} (sin (\frac{1}{x - 1})) = - \frac{cos ( \frac{1}{x - 1} )}{( x - 1 ) ^{2}}

= \frac{1}{2} (3 (x - 1)^{2} sin (\frac{1}{x - 1}) + (- \frac{cos ( \frac{1}{x - 1} )}{( x - 1 ) ^{2}}) (x - 1)^{3})

Simplify

\frac{1}{2} (3 (x - 1)^{2} sin (\frac{1}{x - 1}) + (- \frac{cos ( \frac{1}{x - 1} )}{( x - 1 ) ^{2}}) (x - 1)^{3}) : \frac{1}{2} (3 sin (\frac{1}{x - 1}) (x - 1)^{2} - cos (\frac{1}{x - 1}) (x - 1))

= \frac{1}{2} (3 sin (\frac{1}{x - 1}) (x - 1)^{2} - cos (\frac{1}{x - 1}) (x - 1))

f (x) = \frac{x}{x ^{2} + 16}, at x = 0

\int \frac{x ^{2}}{( 15 + 4 x - 4 x ^{2} ) ^{\frac{3}{2}}} d x

\int (x^{2} e^{x^{3}} + x^{2} sin (y)) d x

\int (\frac{x - 5}{x ^{2} - x - 6}) d x

\int 7 sin (x) d x

What is the derivative of 1/2 (x-1^3sin(1/(x-1))) ?
The derivative of 1/2 (x-1^3sin(1/(x-1))) is 1/2 (3sin(1/(x-1))(x-1)^2-cos(1/(x-1))(x-1))
What is the first derivative of 1/2 (x-1^3sin(1/(x-1))) ?
The first derivative of 1/2 (x-1^3sin(1/(x-1))) is 1/2 (3sin(1/(x-1))(x-1)^2-cos(1/(x-1))(x-1))