Use the fundamental definition of a derivative to find f′(x) where f(x)=x+ax+b . Show all your work. If limit notation is needed in your response, use the following example as a guide: limx→1(x2+2) should be written as lim x→1 (x2+2) .
The fundamental definition of a derivative states that the derivative of a function f(x) at a specific point x is given by the limit of the difference quotient as h approaches 0:
f′(x) = limh→0 (f(x+h) - f(x))/h
Now let's find f′(x) for the function f(x) = x + ax + b:
f′(x) = limh→0 (f(x+h) - f(x))/h
= limh→0 ((x+h) + a(x+h) + b - (x + ax + b))/h
= limh→0 (x + h + ax + ah + b - x - ax - b)/h
= limh→0 (h + ah)/h
= limh→0 (h(1 + a))/h
= limh→0 (1 + a)
= 1 + a
Therefore, f′(x) = 1 + a.
To find the derivative of the function f(x) = x + ax + b using the fundamental definition of a derivative, we need to find the limit of the difference quotient as h approaches 0.
The formula for the difference quotient is:
f'(x) = lim h→0 [f(x + h) - f(x)] / h
Now, let's substitute the expression for f(x):
f'(x) = lim h→0 [(x + h + a(x + h) + b) - (x + ax + b)] / h
Simplify the equation inside the limit:
f'(x) = lim h→0 [x + h + ax + ah + b - x - ax - b] / h
Now, combine like terms:
f'(x) = lim h→0 [h + ah] / h
Factor out the common term:
f'(x) = lim h→0 h(1 + a) / h
Cancel out the h terms:
f'(x) = lim h→0 (1 + a)
Now, take the limit as h approaches 0:
f'(x) = 1 + a
Therefore, the derivative of f(x) = x + ax + b is f'(x) = 1 + a.
Note: In this case, we didn't have to use limit notation since the expression simplified to a constant value.
To find f′(x) using the fundamental definition of a derivative, we need to calculate the limit of the difference quotient as h approaches 0.
The fundamental definition of a derivative is given by:
f′(x) = lim h→0 [f(x + h) - f(x)] / h
Let's proceed to find f′(x) step-by-step for the function f(x) = x + ax + b:
Step 1: Substitute f(x+h) and f(x) into the formula:
f′(x) = lim h→0 [(x + h + a(x + h) + b) - (x + ax + b)] / h
Step 2: Simplify the numerator by distributing a to (x + h):
f′(x) = lim h→0 [(x + h + ax + ah + b) - (x + ax + b)] / h
Simplify further:
f′(x) = lim h→0 [x + h + ax + ah + b - x - ax - b] / h
Step 3: Cancel out like terms:
f′(x) = lim h→0 [(h + ah) / h]
Step 4: Factor out h:
f′(x) = lim h→0 [h(1 + a) / h]
Step 5: Cancel out h in the numerator and denominator:
f′(x) = lim h→0 (1 + a)
Step 6: Evaluate the limit as h approaches 0:
f′(x) = 1 + a
So, the derivative of f(x) = x + ax + b with respect to x is f′(x) = 1 + a.