What is the d/(dt)((d/(dt)(e^tcos(t)))^2) ?

The d/(dt)((d/(dt)(e^tcos(t)))^2) is -4e^tsin(t)(e^tcos(t)-e^tsin(t))

What is the first d/(dt)((d/(dt)(e^tcos(t)))^2) ?

The first d/(dt)((d/(dt)(e^tcos(t)))^2) is -4e^tsin(t)(e^tcos(t)-e^tsin(t))

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\frac{d}{d t} ((\frac{d}{d t} (e^{t} cos (t)))^{2})

- 4 e^{t} sin (t) (e^{t} cos (t) - e^{t} sin (t))

Solution steps

\frac{d}{d t} ((\frac{d}{d t} (e^{t} cos (t)))^{2})

Apply the chain rule:

2 \frac{d}{d t} (e^{t} cos (t)) \frac{d}{d t} (\frac{d}{d t} (e^{t} cos (t)))

= 2 \frac{d}{d t} (e^{t} cos (t)) \frac{d}{d t} (\frac{d}{d t} (e^{t} cos (t)))

\frac{d}{d t} (\frac{d}{d t} (e^{t} cos (t))) = - 2 e^{t} sin (t)

= 2 \frac{d}{d t} (e^{t} cos (t)) (- 2 e^{t} sin (t))

Apply the Product Rule:

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

f = e^{t}, g = cos (t)

= 2 (\frac{d}{d t} (e^{t}) cos (t) + \frac{d}{d t} (cos (t)) e^{t}) (- 2 e^{t} sin (t))

\frac{d}{d t} (e^{t}) = e^{t}

\frac{d}{d t} (cos (t)) = - sin (t)

= 2 (e^{t} cos (t) + (- sin (t)) e^{t}) (- 2 e^{t} sin (t))

Simplify

= - 4 e^{t} sin (t) (e^{t} cos (t) - e^{t} sin (t))

f (x) = \frac{e ^{x^{2}}}{x ^{3}}

x \to - 3 lim (2 x^{2} - 3 x + 4)

\frac{d}{d x} (x^{3} - 9 x^{2} + 24 x - 15)

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

\int \frac{u}{1 + u ^{2}} d u

What is the d/(dt)((d/(dt)(e^tcos(t)))^2) ?
The d/(dt)((d/(dt)(e^tcos(t)))^2) is -4e^tsin(t)(e^tcos(t)-e^tsin(t))
What is the first d/(dt)((d/(dt)(e^tcos(t)))^2) ?
The first d/(dt)((d/(dt)(e^tcos(t)))^2) is -4e^tsin(t)(e^tcos(t)-e^tsin(t))