find the derivative at a given point (by definition)
x^2-6x-16 at x=-2
by definition, the derivative of f(x) is the limit of
(f(x+h)-f(x))/h
So, plug that in for your function and
f(x+h)-f(x) = (x+h)^2 - 6(x+h) - 16 -(x^2-6x-16)
= x^2+2hx+h^2-6x-6h-16-x^2+6x+16
= 2hx+h^2-6h
Now divide that by h, and you get
(f(x+h)-f(x))/h = 2x+h-6
Take the limit, and df/dx = 2x-6
at x = -2, df/dx = -18
What does df/do mean?
Typo dx
To find the derivative of a function at a given point by definition, we can start by using the definition of the derivative:
f'(a) = lim(h->0) [f(a+h) - f(a)] / h
In this case, we want to find the derivative of the function f(x) = x^2 - 6x - 16 at the point x = -2. Let's substitute these values into the definition of the derivative equation:
f'(-2) = lim(h->0) [f(-2+h) - f(-2)] / h
Now, we'll substitute the function values and simplify the equation:
f'(-2) = lim(h->0) [(-2+h)^2 - 6(-2+h) - 16 - (-2)^2 + 6(-2) + 16] / h
= lim(h->0) [(h^2 + 4h + 4) - (-12 + 6h) - 16 - 4 + 12 + 16] / h
= lim(h->0) [h^2 + 4h + 4 + 12 - 6h - 16 - 4 + 12 + 16] / h
= lim(h->0) [h^2 - 2h + 12] / h
Next, we can simplify the numerator and divide every term by h:
f'(-2) = lim(h->0) h^2 / h - 2h / h + 12 / h
= lim(h->0) h - 2 + 12 / h
Now, we can evaluate the limit as h approaches 0:
f'(-2) = (0) - 2 + 12 / 0
= -2 + 12 / 0
Since the denominator is zero, this limit is undefined. Therefore, the derivative of the function f(x) = x^2 - 6x - 16 at the point x = -2 does not exist.