# First derivatives.Find the first derivative of v(t) = ( 1 + 3^t) / 3^t.

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We need the first derivative of v(t) = ( 1 + 3^t) / 3^t

v(t) = ( 1 + 3^t) / 3^t

=> 1/3^t + 1

=> (1/3)^t + 1

v'(t) = log (1/3) *(1/3)^t

**The required first derivative of v(t) = ( 1 + 3^t) / 3^t is log(1/3)*(1/3)^t**

We'll differentiate the given function, with respect to t.

We'll use the quotient rule:

v'(t) = [(1+3^t)'*(3^t) - (1+3^t)*(3^t)']/(3^t)^2

We'll differentiate and we'll get:

v'(t) = [(3^t*ln3)*(3^t) - (3^t*ln3)*(1+3^t)]/(3^t)^2

v'(t) = [(3^t*ln3)*(3^t - 1 -3^t)]/(3^t)^2

We'll eliminate like terms from numerator:

v'(t) = -(3^t*ln3)/(3^t)^2

We'll simplify and we'll get:

v'(t) = -(ln3)/(3^t)

v'(t) = (ln 1/3)/(3^t)

**The first derivative of v(t)=(1+3^t)/3^t is:**

**v'(t) = (ln 1/3)/(3^t)**