What is the derivative of (sinh(x)^{2x^2+2x}) ?

The derivative of (sinh(x)^{2x^2+2x}) is sinh^{2x^2+2x}(x)(ln(sinh(x))(4x+2)+coth(x)(2x^2+2x))

What is the first derivative of (sinh(x)^{2x^2+2x}) ?

The first derivative of (sinh(x)^{2x^2+2x}) is sinh^{2x^2+2x}(x)(ln(sinh(x))(4x+2)+coth(x)(2x^2+2x))

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\frac{d}{d x} ((sinh (x))^{2 x^{2} + 2 x})

sinh^{2 x^{2} + 2 x} (x) (ln (sinh (x)) (4 x + 2) + coth (x) (2 x^{2} + 2 x))

Solution steps

\frac{d}{d x} ((sinh (x))^{2 x^{2} + 2 x})

Apply exponent rule:

a^{b} = e^{b l n (a)}

(sinh (x))^{2 x^{2} + 2 x} = e^{(2 x^{2} + 2 x) l n (s i n h (x))}

= \frac{d}{d x} (e^{(2 x^{2} + 2 x) l n (s i n h (x))})

Apply the chain rule:

e^{(2 x^{2} + 2 x) l n (s i n h (x))} \frac{d}{d x} ((2 x^{2} + 2 x) ln (sinh (x)))

= e^{(2 x^{2} + 2 x) l n (s i n h (x))} \frac{d}{d x} ((2 x^{2} + 2 x) ln (sinh (x)))

\frac{d}{d x} ((2 x^{2} + 2 x) ln (sinh (x))) = (4 x + 2) ln (sinh (x)) + coth (x) (2 x^{2} + 2 x)

= e^{(2 x^{2} + 2 x) l n (s i n h (x))} ((4 x + 2) ln (sinh (x)) + coth (x) (2 x^{2} + 2 x))

e^{(2 x^{2} + 2 x) l n (s i n h (x))} = sinh^{2 x^{2} + 2 x} (x)

= sinh^{2 x^{2} + 2 x} (x) (ln (sinh (x)) (4 x + 2) + coth (x) (2 x^{2} + 2 x))

\int \frac{( ln ( x ) ) ^{2}}{2} d x

y^{^{'}} = 6 y^{2} + x y^{2}, y (0) = 1

\frac{d ^{2}}{d x ^{2}} (\frac{2000 t}{4 t + 75})

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

x y = 2

What is the derivative of (sinh(x)^{2x^2+2x}) ?
The derivative of (sinh(x)^{2x^2+2x}) is sinh^{2x^2+2x}(x)(ln(sinh(x))(4x+2)+coth(x)(2x^2+2x))
What is the first derivative of (sinh(x)^{2x^2+2x}) ?
The first derivative of (sinh(x)^{2x^2+2x}) is sinh^{2x^2+2x}(x)(ln(sinh(x))(4x+2)+coth(x)(2x^2+2x))