Question

Differentiate.

\(g\left( t \right) = \frac{{3 - 2t}}{{5t + 1}}\)

Short answer

Derivative of the function is \( - \frac{{17}}{{{{\left( {5t + 1} \right)}^2}}}\).

Step-by-step solution

Precise Definition of differentiation

A derivative is a function of a real variable. It is the rate of change of output value with respect to an input value. The derivative function is a single variable at a selected input value.

Find the derivative of g using the Quotient rule

The equation for the Quotient ruleis \({\left( {\frac{f}{g}} \right)^\prime } = \frac{{gf' - fg'}}{{{{\left( g \right)}^2}}}\)

Apply Quotient rule for the function \(g(t) = \frac{{3 - 2t}}{{5t + 1}}\).

\(\frac{d}{{dt}}\left( {g\left( t \right)} \right) = \frac{d}{{dt}}\left( {\frac{{3 - 2t}}{{5t + 1}}} \right)\)

Simplify the obtained condition

Apply derivative to obtain the answer.

\(\begin{aligned}g'(t) &= \frac{{\left( {5t + 1} \right)\frac{d}{{dt}}\left( {3 - 2t} \right) - \left( {3 - 2t} \right)\frac{d}{{dt}}\left( {5t + 1} \right)}}{{{{\left( {5t + 1} \right)}^2}}}\\ &= \frac{{\left( {5t + 1} \right)\left( { - 2} \right) - \left( {3 - 2t} \right)\left( 5 \right)}}{{{{\left( {5t + 1} \right)}^2}}}\\ &= \frac{{ - 10t - 2 - 15 + 10t}}{{{{\left( {5t + 1} \right)}^2}}}\\ &= - \frac{{17}}{{{{\left( {5t + 1} \right)}^2}}}\end{aligned}\)

Thus, the derivative of \(g\left( t \right)\) is \( - \frac{{17}}{{{{\left( {5t + 1} \right)}^2}}}\).