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Question 06 

FullSolution Problems. In questions 28, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested. Find the derivative of the function using the definition of the derivative. NOTE: No credit will be given for using differentiation rules such as the power rule, product rule, quotient rule or chain rule, although you may use these rules to check your answer. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The definition of a derivative is
Alternatively, one can use

Hint 2 

At some stage, you might want to find a common denominator for two fractions you are trying to subtract. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. By the definition of a derivative, we have that
The key step to simplifying this expression is here: we find a common denominator for the fractions on top and combine them.
Now we simplify the expression into a single fraction and cancel the h on top and bottom.
Finally taking the limit gives:

Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Using the alternative definition,
Here we find a common denominator to simplify the fractions on top of the large fraction.
We then simplify the fraction, factoring out a 5 on top and then canceling the (x  t) term.
Finally we take the limit.
Which completes the question. 