Final answer:
a. f'(x) = 4x, b. g'(x) = 2/x - 2x, c. h'(x) = √3
Step-by-step explanation:
a. Derivative of f(x) = 2x²-3x+4
To find the derivative of f(x), we use the limit definition of the derivative. The limit definition is given as:
f'(x) = lim(h->0) (f(x+h) - f(x))/h
Substituting the given function into the limit definition:
f'(x) = lim(h->0) ((2(x+h)² - 3(x+h) + 4) - (2x² - 3x + 4))/h
Simplifying the expression:
f'(x) = lim(h->0) (2h² + 4xh)/(h)
Canceling out h:
f'(x) = lim(h->0) (2h + 4x)
Taking the limit as h approaches 0:
f'(x) = 4x
b. Derivative of g(x) = 1/x² + 1
To find the derivative of g(x), we again use the limit definition of the derivative:
g'(x) = lim(h->0) (g(x+h) - g(x))/h
Substituting the given function:
g'(x) = lim(h->0) (1/(x+h)² + 1 - (1/x² + 1))/h
Simplifying the expression:
g'(x) = lim(h->0) ((x² + xh + h² - x² - x²h - h² + x² + 2x + 1) - (1 + x²))/h
Canceling out h:
g'(x) = lim(h->0) (xh - x²h + 2x + 1)/h
Taking the limit as h approaches 0:
g'(x) = 2/x - 2x
c. Derivative of h(x) = √3x-2
Using the limit definition of the derivative:
h'(x) = lim(h->0) (h(x+h) - h(x))/h
Substituting the given function:
h'(x) = lim(h->0) (√3(x+h)-2 - (√3x-2))/h
Simplifying the expression:
h'(x) = lim(h->0) (√3x+√3h-2 - √3x+2)/h
Canceling out h:
h'(x) = lim(h->0) (√3h)/h
Taking the limit as h approaches 0:
h'(x) = √3