Questions: f(t)=6/t^6-4/t^4+3/t f'(t)=□

Solution Steps

To find the derivative \( f'(t) \) of the function \( f(t) = \frac{6}{t^6} - \frac{4}{t^4} + \frac{3}{t} \), we will apply the power rule for differentiation. Rewrite each term with negative exponents, differentiate each term separately, and then combine the results.

The given function is \( f(t) = \frac{6}{t^6} - \frac{4}{t^4} + \frac{3}{t} \). We can rewrite this function using negative exponents as follows: \[ f(t) = 6t^{-6} - 4t^{-4} + 3t^{-1} \]

To find the derivative \( f'(t) \), we apply the power rule to each term. The power rule states that the derivative of \( t^n \) is \( nt^{n-1} \).

• The derivative of \( 6t^{-6} \) is \( -36t^{-7} \).
• The derivative of \( -4t^{-4} \) is \( 16t^{-5} \).
• The derivative of \( 3t^{-1} \) is \( -3t^{-2} \).

Combine the derivatives of each term to find \( f'(t) \): \[ f'(t) = -36t^{-7} + 16t^{-5} - 3t^{-2} \]

Convert the negative exponents back to fractions: \[ f'(t) = -\frac{36}{t^7} + \frac{16}{t^5} - \frac{3}{t^2} \]

Final Answer