Find the indicated derivative. d / d t ( 5 − 7 t − 3 ) d/dt(5-7t^{-3})

In this problem, we're asked to find the derivative d d t of 5 minus 7 t to the power of negative 3. So let's go ahead and jump into this derivative here. Now here, d d t tells us we're differentiating with respect to t, which makes sense because in our function here, we have the variable t. So let's go ahead and break this down. Now the derivative, d dt of 5 minus 7 t to the power of negative 3, I can break this down in each of my terms because this means that I'm taking the derivative d dt of 5, and I am then subtracting the derivative of this second term here. Since I have a constant, I can go ahead and pull that constant out to the front. So this is effectively a negative 7 times d dt of t to the power of negative 3. Now looking at this, my very first term, d dt of 5, this is a constant, meaning that my derivative here is just 0. So I have 0 minus 7 times the derivative of t to the power of negative 3. Now even though I have a negative exponent here, I can still, of course, use the power rule no matter what my power is. So using my power rule, I can multiply by that negative 3 and then decrease that exponent by 1, giving me negative 3 times t to the power of negative 3 minus 1, which is negative 4. Now I can further simplify this because this 0 is not doing anything here, and I can go ahead and multiply my negative 7 7 times negative 3 to give me a positive 21. So this is positive 21 times t to the power of negative 4. Now you may also see this rewritten as 21 over t to the power of 4 because remember that whenever you have a negative exponent, that just means that it's on the bottom of a fraction. So either of these answers are correct here for the derivative of our function, 5 minus 7 t to the power of negative 3. Let me know if you have any questions, and I'll see you in the next one.

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m