Certainly! Let's find the derivative of the function [tex]\( f(t) = \frac{7}{t^7} - \frac{5}{t^5} + \frac{4}{t} \)[/tex].
First, we will rewrite each term of the function [tex]\( f(t) \)[/tex] as a power of [tex]\( t \)[/tex]:
[tex]\[ \frac{7}{t^7} = 7t^{-7} \][/tex]
[tex]\[ \frac{5}{t^5} = 5t^{-5} \][/tex]
[tex]\[ \frac{4}{t} = 4t^{-1} \][/tex]
Now, we can express [tex]\( f(t) \)[/tex] as:
[tex]\[ f(t) = 7t^{-7} - 5t^{-5} + 4t^{-1} \][/tex]
The next step is to differentiate each term individually using the power rule of differentiation, which states that if [tex]\( f(t) = t^n \)[/tex], then [tex]\( f'(t) = nt^{n-1} \)[/tex].
1. Differentiating [tex]\( 7t^{-7} \)[/tex]:
[tex]\[ \frac{d}{dt}(7t^{-7}) = 7 \cdot (-7)t^{-8} = -49t^{-8} \][/tex]
2. Differentiating [tex]\( 5t^{-5} \)[/tex]:
[tex]\[ \frac{d}{dt}(5t^{-5}) = 5 \cdot (-5)t^{-6} = -25t^{-6} \][/tex]
3. Differentiating [tex]\( 4t^{-1} \)[/tex]:
[tex]\[ \frac{d}{dt}(4t^{-1}) = 4 \cdot (-1)t^{-2} = -4t^{-2} \][/tex]
Finally, combining these results, we have:
[tex]\[ f'(t) = -49t^{-8} - 25t^{-6} - 4t^{-2} \][/tex]
We can rewrite this in terms of negative exponents back to fractions:
[tex]\[ f'(t) = -\frac{49}{t^8} + \frac{25}{t^6} - \frac{4}{t^2} \][/tex]
Thus, the derivative of the function is:
[tex]\[
f^{\prime}(t) = -\frac{49}{t^8} + \frac{25}{t^6} - \frac{4}{t^2}
\][/tex]
This is the final answer.
