Estimate the indicated derivative by any method
y=1-x^2 ; estimate dy/dx , x=-1
I know I would use the f(x)= f(x+H)-(x)/h I am a bit overwhelmed with what to do next. Thanks John
To estimate the derivative dy/dx of the function y = 1 - x^2 at x = -1, you can use the method you mentioned, called the difference quotient. Here's how you can proceed step by step:
1. Start with the equation for the difference quotient:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
2. Plug in the given function y = 1 - x^2 into the difference quotient:
f'(x) = lim(h->0) [(1 - (x + h)^2) - (1 - x^2)] / h
3. Simplify the expression inside the limit:
f'(x) = lim(h->0) [(1 - (x^2 + 2xh + h^2)) - (1 - x^2)] / h
= lim(h->0) [(1 - x^2 - 2xh - h^2) - (1 - x^2)] / h
= lim(h->0) [-2xh - h^2] / h
4. Cancel out the common factor of h in the numerator:
f'(x) = lim(h->0) [-2x - h] = -2x
5. Substitute x = -1 into the expression to find the estimated derivative at x = -1:
f'(-1) = -2*(-1) = 2
So, the estimated derivative dy/dx of the function y = 1 - x^2 at x = -1 is 2.