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Popular >

derivative of arctan(tanh(x))

Solution

\frac{d}{d x} (arctan (tanh (x)))

Solution

\frac{sech ^{2} ( x )}{tanh ^{2} ( x ) + 1}

Solution steps

\frac{d}{d x} (arctan (tanh (x)))

Apply the chain rule:

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

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

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

= \frac{1}{( tanh ( x ) ) ^{2} + 1} sech^{2} (x)

Simplify

\frac{1}{tanh ^{2} ( x ) + 1} sech^{2} (x) : \frac{sech ^{2} ( x )}{tanh ^{2} ( x ) + 1}

= \frac{sech ^{2} ( x )}{tanh ^{2} ( x ) + 1}

Popular Examples

integral of (2x^3+1)/(x^2+x-6)

\int \frac{2 x ^{3} + 1}{x ^{2} + x - 6} d x

integral of 1/(sqrt(2x^2-4x+6))

\int \frac{1}{2 x ^{2} - 4 x + 6} d x

integral from 0 to 1 of 2pix(33-33x)

\int_{0}^{1} 2 π x (33 - 33 x) d x

limit as x approaches 0 of (4/(x+4)-1)/x

x \to 0 lim (\frac{\frac{4}{x + 4} - 1}{x})

integral from 0 to 1 of (73)/(x^5)

\int_{0}^{1} \frac{73}{x ^{5}} d x

Frequently Asked Questions (FAQ)

What is the derivative of arctan(tanh(x)) ?
The derivative of arctan(tanh(x)) is (sech^2(x))/(tanh^2(x)+1)
What is the first derivative of arctan(tanh(x)) ?
The first derivative of arctan(tanh(x)) is (sech^2(x))/(tanh^2(x)+1)