What is the d/(dt)((A(t))(12-(5A(t))/(500-t))) ?

The d/(dt)((A(t))(12-(5A(t))/(500-t))) is A((5At^2-5000At)/((-t+500)^2)+12)

What is the first d/(dt)((A(t))(12-(5A(t))/(500-t))) ?

The first d/(dt)((A(t))(12-(5A(t))/(500-t))) is A((5At^2-5000At)/((-t+500)^2)+12)

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\frac{d}{d t} ((A (t)) (12 - \frac{5 A ( t )}{500 - t}))

A (\frac{5 A t ^{2} - 5000 A t}{( - t + 500 ) ^{2}} + 12)

Solution steps

\frac{d}{d t} ((A (t)) (12 - \frac{5 A ( t )}{500 - t}))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= A \frac{d}{d t} (t (12 - \frac{5 A t}{500 - t}))

Apply the Product Rule:

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

f = t, g = 12 - \frac{5 A t}{500 - t}

= A (\frac{d t}{d t} (12 - \frac{5 A t}{500 - t}) + \frac{d}{d t} (12 - \frac{5 A t}{500 - t}) t)

\frac{d t}{d t} = 1

\frac{d}{d t} (12 - \frac{5 A t}{500 - t}) = - \frac{2500 A}{( - t + 500 ) ^{2}}

= A (1 \cdot (12 - \frac{5 A t}{500 - t}) + (- \frac{2500 A}{( - t + 500 ) ^{2}}) t)

Simplify

A (1 \cdot (12 - \frac{5 A t}{500 - t}) + (- \frac{2500 A}{( - t + 500 ) ^{2}}) t) : A (\frac{5 A t ^{2} - 5000 A t}{( - t + 500 ) ^{2}} + 12)

= A (\frac{5 A t ^{2} - 5000 A t}{( - t + 500 ) ^{2}} + 12)

\frac{( x + 5 )}{( 3 - x ) ( 1 + x ) ^{2}}

(\frac{a + 1}{a}) 5

\frac{6}{( x + 1 ) ^{3}}

(x^{2} - 2 x + 3) d x

What is the d/(dt)((A(t))(12-(5A(t))/(500-t))) ?
The d/(dt)((A(t))(12-(5A(t))/(500-t))) is A((5At^2-5000At)/((-t+500)^2)+12)
What is the first d/(dt)((A(t))(12-(5A(t))/(500-t))) ?
The first d/(dt)((A(t))(12-(5A(t))/(500-t))) is A((5At^2-5000At)/((-t+500)^2)+12)