How do you differentiate #s=(t^2-1)(t^2+1)#?

Question

Emma Aldridge · Accepted Answer

# 4 t^3 # First you open the bracket and multiply the two terms, #s=(t^2-1)(t^2+1)# # s= t^4 + t^2 - t^2 - 1 # # s = t^4 - 1 # Now, differentiate for t. #(ds)/(dt)# = # 4 * t ^ (4-1) - 0 # = # 4 t^3 #

Christopher Allers · Answer

undefined To differentiate $ s = (t^2 - 1)(t^2 + 1) $:

1. Recognize that $ s $ is a function of $ t $, so you'll differentiate with respect to $ t $.

2. Use the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

3. Identify the two functions being multiplied: $ u(t) = t^2 - 1 $ and $ v(t) = t^2 + 1 $.

4. Find the derivatives of $ u(t) $ and $ v(t) $ with respect to $ t $:
   - $ u'(t) = 2t $ (since the derivative of $ t^2 - 1 $ with respect to $ t $ is $ 2t $)
   - $ v'(t) = 2t $ (since the derivative of $ t^2 + 1 $ with respect to $ t $ is $ 2t $)

5. Apply the product rule:
   $$ s'(t) = u'(t)v(t) + u(t)v'(t) $$

6. Plug in the derivatives and original functions:
   $$ s'(t) = (2t)(t^2 + 1) + (t^2 - 1)(2t) $$

7. Simplify the expression:
   $$ s'(t) = 2t^3 + 2t + 2t^3 - 2t $$

8. Combine like terms:
   $$ s'(t) = 4t^3 $$