$$ \frac { d } { d t } ( a t ^ { 5 } - s b t ^ { 3 } ) $$

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The derivative of $$ a t^5 - s b t^3 $$ with respect to $$ t $$ is $$ 5a t^4 - 3s b t^2 $$. To differentiate the expression $$ a t^5 - s b t^3 $$ with respect to $$ t $$, we will apply the power rule of differentiation. The power rule states that if $$ f(t) = t^n $$, then $$ \frac{d}{dt} f(t) = n t^{n-1} $$. Let's differentiate each term in the expression: 1. For the term $$ a t^5 $$: $$ \frac{d}{dt}(a t^5) = a \cdot 5 t^{5-1} = 5a t^4 $$ 2. For the term $$ -s b t^3 $$: $$ \frac{d}{dt}(-s b t^3) = -s b \cdot 3 t^{3-1} = -3s b t^2 $$ Now, we can combine the results of the differentiation: $$ \frac{d}{dt}(a t^5 - s b t^3) = 5a t^4 - 3s b t^2 $$ Thus, the derivative is: $$ \frac{d}{dt}(a t^5 - s b t^3) = 5a t^4 - 3s b t^2 $$

\( \frac { d } { d t } ( a t ^ { 5 } - s b t ^ { 3 } ) \)

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\( \frac { d } { d t } ( a t ^ { 5 } - s b t ^ { 3 } ) \)

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