find the derivative of f(x)=3x^2-x using the limit definition.
f(x) = 3x^2 - x
f(x+h) = 3(x+h)^2 - (x+h)
= 3x^2 + 6xh + 3h^2 - x - h
dy/dx = lim [ ( f(x+h) - f(x)]/h , as h ----> 0
= lim (3x^2 + 6xh + 3h^2 - x - h - 3x^2 + x)/h
= lim (6xh + 3h^2 - h)/h
= lim 6x + 3h - 1 , as h --->0
= 6x - 1
(the h--> 0 should be under the word "lim" in each line)
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To find the derivative of a function using the limit definition, we need to evaluate the following limit:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Let's substitute the function f(x) = 3x^2 - x into this expression:
f'(x) = lim(h->0) [ (3(x + h)^2 - (x + h)) - (3x^2 - x) ] / h
Now, expand and simplify the expression inside the limit:
f'(x) = lim(h->0) [ (3x^2 + 6xh + 3h^2 - x - h) - (3x^2 - x) ] / h
f'(x) = lim(h->0) [ 3x^2 + 6xh + 3h^2 - x - h - 3x^2 + x ] / h
f'(x) = lim(h->0) [ 6xh + 3h^2 - h ] / h
Now, we can cancel out the common factor of h in the numerator:
f'(x) = lim(h->0) 6x + 3h - 1
Finally, take the limit as h approaches 0:
f'(x) = 6x - 1
Therefore, the derivative of f(x) = 3x^2 - x using the limit definition is f'(x) = 6x - 1.
To find the derivative of a function using the limit definition, you need to compute the following limit:
f'(x) = lim (h->0) [f(x+h) - f(x)] / h
Let's substitute the given function, f(x) = 3x^2 - x, into the formula:
f'(x) = lim (h->0) [(3(x+h)^2 - (x+h)) - (3x^2 - x)] / h
Now we can expand and simplify:
f'(x) = lim (h->0) [(3(x^2 + 2xh + h^2) - (x + h)) - (3x^2 - x)] / h
= lim (h->0) [3x^2 + 6xh + 3h^2 - x - h - 3x^2 + x] / h
= lim (h->0) (6xh + 3h^2 - h) / h
= lim (h->0) h(6x + 3h - 1) / h
Now, we can cancel out the factor of h:
f'(x) = lim (h->0) 6x + 3h - 1
= 6x + 0 - 1
= 6x - 1
Therefore, the derivative of f(x) = 3x^2 - x using the limit definition is f'(x) = 6x - 1.