Questions: Differentiate the function. V(t)=t^-3 / 5+t^7 V'(t)=

Solution Steps

To differentiate the function \( V(t) = t^{-3/5} + t^7 \), apply the power rule for differentiation. The power rule states that the derivative of \( t^n \) is \( n \cdot t^{n-1} \). Apply this rule to each term in the function separately.

To differentiate the function \( V(t) = t^{-3/5} + t^7 \), apply the power rule to each term. The power rule states that the derivative of \( t^n \) is \( n \cdot t^{n-1} \).

• For the term \( t^{-3/5} \), the derivative is: \[ \frac{d}{dt} \left( t^{-3/5} \right) = -\frac{3}{5} \cdot t^{-3/5 - 1} = -\frac{3}{5} \cdot t^{-8/5} \]

• For the term \( t^7 \), the derivative is: \[ \frac{d}{dt} \left( t^7 \right) = 7 \cdot t^{7-1} = 7 \cdot t^6 \]

Combine the derivatives of each term to find the derivative of the entire function: \[ V'(t) = -\frac{3}{5} \cdot t^{-8/5} + 7 \cdot t^6 \]

Express the derivative in a simplified form: \[ V'(t) = -\frac{0.6}{t^{1.6}} + 7t^6 \]

Final Answer