What is the derivative of arctan(sin(x^2+pi/x)) ?

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

What is the first derivative of arctan(sin(x^2+pi/x)) ?

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

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\frac{d}{d x} (arctan (sin (x^{2} + \frac{π}{x})))

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

Solution steps

\frac{d}{d x} (arctan (sin (x^{2} + \frac{π}{x})))

Apply the chain rule:

\frac{1}{( sin ( x ^{2} + \frac{π}{x} ) ) ^{2} + 1} \frac{d}{d x} (sin (x^{2} + \frac{π}{x}))

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

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

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

Simplify

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

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

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