What is the first d/(dt)(1-cos(t)) ?

The first d/(dt)(1-cos(t)) is sin(t)

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d/(dt)(1-cos(t))

\frac{d}{d t} (1 - cos (t))

sin (t)

Solution steps

\frac{d}{d t} (1 - cos (t))

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d t} (1) - \frac{d}{d t} (cos (t))

\frac{d}{d t} (1) = 0

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

= 0 - (- sin (t))

Simplify

= sin (t)

\frac{d}{d x} (cos^{2} (5 - x))

\frac{d y}{d t} + e^{t} y = - 5 e^{t}

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

\int (\frac{1}{( 25 - x ^{2} ) ^{\frac{3}{2}}}) d x

\int \frac{5 + x}{5 - x} d x