Question

find the derivative of the function.

s = 8 t ^ { 7 / 4 } + 4 e ^ { 2 }

\frac { d s } { d t } =

Explanation:

Step1: Differentiate the first term

Use the power rule \(\frac{d}{dt}(at^n)=ant^{n - 1}\). For \(y = 8t^{7/4}\), \(a = 8\), \(n=\frac{7}{4}\). Then \(\frac{d}{dt}(8t^{7/4})=8\times\frac{7}{4}t^{\frac{7}{4}-1}\).
Simplify \(8\times\frac{7}{4}t^{\frac{7}{4}-1}=14t^{3/4}\).

Step2: Differentiate the second term

Since \(4e^{2}\) is a constant (because \(e^{2}\approx7.39\) is a constant and multiplying by 4 still gives a constant). The derivative of a constant \(C\) with respect to \(t\) is \(0\), so \(\frac{d}{dt}(4e^{2}) = 0\).

Step3: Combine the derivatives

By the sum - rule of differentiation \(\frac{d}{dt}(u + v)=\frac{du}{dt}+\frac{dv}{dt}\), where \(u = 8t^{7/4}\) and \(v = 4e^{2}\). So \(\frac{ds}{dt}=\frac{d}{dt}(8t^{7/4})+\frac{d}{dt}(4e^{2})\).

Answer:

\(14t^{3/4}\)