You are given the following function.
f(x)= (8+x)/(1-8x)
(a) Find the derivative of the function using the definition of derivative.
(b) Give the domain of the function.
c) Give the domain of derivative of function?
use the quotient rule
dy/dx = [(1-8x)(1) - (8+x)(-8)]/(1-8x)^2
= ... (you simplify it)
b) domain of f(x) is any real number, x not equal to 1/8
c) same as b)
To find the derivative of a function using the definition of derivative, we will use the limit definition of the derivative. The derivative of a function f(x) is defined as:
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
(a) Let's find the derivative of the given function f(x) using the definition of derivative:
f(x) = (8+x) / (1 - 8x)
Using the definition of derivative, we have:
f'(x) = lim(h→0) [(8+(x+h))/(1-8(x+h)) - (8+x)/(1-8x)] / h
Simplifying this expression, we get:
f'(x) = lim(h→0) [(8+x+h)(1-8x) - (8+x)(1-8(x+h))] / [(1-8(x+h))(1-8x)] / h
Expanding and rearranging terms, we have:
f'(x) = lim(h→0) [8 + x + h - 64x - 8h + 8x + 64(x+h)] / [(1-8(x+h))(1-8x)] / h
Now, let's simplify further:
f'(x) = lim(h→0) [72 + 64h] / [(1-8(x+h))(1-8x)] / h
Taking the limit as h approaches 0, we have:
f'(x) = lim(h→0) (72 + 64h) / [(1-8(x+h))(1-8x)] / h
= (72 + 64(0)) / [(1-8(x+0))(1-8x)]
= 72 / (1-8x)(1-8x)
Therefore, the derivative of the function f(x) = (8+x) / (1 - 8x) using the definition of derivative is f'(x) = 72 / (1-8x)(1-8x).
(b) To find the domain of the function, we need to identify values of x that make the denominator of the function non-zero.
In this case, the function f(x) has a denominator (1-8x) in which it becomes zero when x = 1/8. So, the domain of the function f(x) is all real numbers except x = 1/8.
Therefore, the domain of the function f(x) is (-∞, 1/8) ∪ (1/8, +∞).
(c) The derivative of a function is defined for all values in the domain of the original function. So, the domain of the derivative function f'(x) is the same as the domain of the original function f(x), which is (-∞, 1/8) ∪ (1/8, +∞).