How do you use the limit definition to find the derivative of #f(x)=x^2-15x+7#?
1 Answer
Expand, reduce and evaluate the limit.
Explanation:
I'll use
Long version explanation
For
notice that
# = lim_(hrarr0)(((x+h)^2-15(x+h)+7)-(x^2-15x+7))/h#
Notice that, if we try to evaluate by substitutiton, we get the indeterminate form
Expand the numerator:
(note
# = lim_(hrarr0)((x^2+2xh+h^2-15x-15h+7)-(x^2-15x+7))/h#
# = lim_(hrarr0)((x^2+2xh+h^2-15x-15h+7-x^2+15x-7))/h#
Now, some of the terms in the numerator add to
# = lim_(hrarr0)((color(red)(x^2)+2xh+h^2 color(green)(-15x) -15hcolor(blue)(+7) color(red)(-x^2) color(green)(+15x)color(blue)(-7)))/h#
# = lim_(hrarr0)(2xh+h^2-15h)/h#
We still get
# = lim_(hrarr0)(cancel(h)(2x+h-15))/cancel(h)_1#
# = lim_(hrarr0)(2x+h-15)#
And now we can evaluate the limit
# = 2x+(0)-15 = 2x-15#
So,
Short version
# = lim_(hrarr0)(((x+h)^2-15(x+h)+7)-(x^2-15x+7))/h#
# = lim_(hrarr0)((x^2+2xh+h^2-15x-15h+7-x^2+15x-7))/h#
# = lim_(hrarr0)(2xh+h^2-15h)/h#
# = lim_(hrarr0)(2x+h-15)#
# = 2x-15#