What is the derivative of arctan(x+2y) ?

The derivative of arctan(x+2y) is (1+2\frac{dy)/(dx)}{(x+2y)^2+1}

What is the first derivative of arctan(x+2y) ?

The first derivative of arctan(x+2y) is (1+2\frac{dy)/(dx)}{(x+2y)^2+1}

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\frac{d}{d x} (arctan (x + 2 y))

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

Solution steps

\frac{d}{d x} (arctan (x + 2 y))

Apply the chain rule:

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

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

\frac{d}{d x} (x + 2 y) = 1 + 2 \frac{d y}{d x}

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

Simplify

\frac{1}{( x + 2 y ) ^{2} + 1} (1 + 2 \frac{d y}{d x}) : \frac{1 + 2 \frac{d y}{d x}}{( x + 2 y ) ^{2} + 1}

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

What is the derivative of arctan(x+2y) ?
The derivative of arctan(x+2y) is (1+2\frac{dy)/(dx)}{(x+2y)^2+1}
What is the first derivative of arctan(x+2y) ?
The first derivative of arctan(x+2y) is (1+2\frac{dy)/(dx)}{(x+2y)^2+1}