Use the definition of the derivative to find f'(x)
1. f(x) 2x^2 + x - 1
Before the final step, my 'result' looked weird and I think I did something wrong in my arithmetic...someone correct me?
Here's my work...
BEGINNING STEP:
lim [2(x + deltaX)^2 - (x + deltaX) - (2x^2 + x - 1)] / deltaX
near the end step:
lim [4x + 2deltaX - 2x^2 - 1] / deltaX
Thanks
You started correctly, but you must have a math error in the intermediate steps.
((2(x+deltaX)^2 + (x+deltaX) - 1) - (2x^2 + x - 1) )/deltaX
=((2x2+4(x*deltaX) + 2deltaX^2 + x + deltaX - 1) - (2x^2 + x - 1) ) /deltaX
=(4(x*deltaX) + 2*deltaX^2 + deltaX)/deltaX
=4x + deltaX + 1
limit as delatX goes to 0
=4x + 1
Ooops!
the step before the limit statement should have read:
4x + 2deltaX + 1
To find the derivative of the function f(x) = 2x^2 + x - 1 using the definition of the derivative, you're on the right track! It seems that you made a small mistake in expanding the terms.
Let's go through the steps again:
Starting from the beginning step:
lim [2(x + deltaX)^2 - (x + deltaX) - (2x^2 + x - 1)] / deltaX
Expanding the terms, we have:
lim [2(x^2 + 2x*deltaX + (deltaX)^2) - (x + deltaX) - (2x^2 + x - 1)] / deltaX
Simplifying further:
lim [2x^2 + 4x*deltaX + 2(deltaX)^2 - x - deltaX - 2x^2 - x + 1] / deltaX
Combining like terms:
lim [4x*deltaX + 2(deltaX)^2 - deltaX + 1] / deltaX
Factoring out a common factor of deltaX:
lim deltaX[4x + 2(deltaX) - 1] / deltaX
Next, cancel out the deltaX terms:
lim [4x + 2(deltaX) - 1]
Now, we can take the limit as deltaX approaches 0:
4x + 2(0) - 1 = 4x - 1
Therefore, the derivative of f(x) = 2x^2 + x - 1 is f'(x) = 4x - 1.
So, in your final step, it should be f'(x) = 4x - 1 instead of f'(x) = 4x + 2deltaX - 2x^2 - 1.