Question

Find the derivative of the function using the definition of derivative. f(x) = 6 + x 1 − 6x f '(x) = State the domain of the function. (Enter your answer in interval notation.) State the domain of its derivative. (Enter your answer in interval notation.)

Answer 1

The derivative of the corrected function f(x) = 6 - 5x is f'(x) = -5. The domain of both the function and its derivative is all real numbers, expressed in interval notation as (-∞, ∞).

Step-by-step explanation:

The question asks to find the derivative of the function f(x) = 6 + x - 6x using the definition of derivative and to state the domain of the function and its derivative. However, there seems to be a typo in the expression of the function. Assuming the correct expression of the function is f(x) = 6 + x - 6x, which simplifies to f(x) = 6 - 5x, let's compute the derivative:

The derivative of f(x) using the power rule is:

f'(x) = d/dx(6) - 5*d/dx(x) = 0 - 5 = -5

Since the derivative is a constant, the function is a linear function and its graph is a straight line. The domain of a linear function is all real numbers, so the domain of f(x) is (-∞, ∞). Similarly, the domain of its derivative f'(x) is also all real numbers: (-∞, ∞).

Answer 2

`f'(x)=(37)/((1-6x)^2)`

Domain of f(x):
`(-\infty,(1)/(6))\cup ((1)/(6),\infty)`.

Domain of f'(x):
`Domain=(-\infty,(1)/(6))\cup ((1)/(6),\infty)`.

Step-by-step explanation:

Consider the given function

`f(x)=(6+x)/(1-6x)`

Domain of this function is all real numbers except those numbers for which denominator is equal to 0.

`1-6x=0`

`1=6x`

Divide both sides by 6.

`(1)/(6)=x`

The function f(x) is not defined for 1/6. So, the domain of the function f(x) is

`Domain=(-\infty,(1)/(6))\cup ((1)/(6),\infty)`

Find the derivative of the function using the definition of derivative.

`f'(x)=lim_(x\rightarrow 0)(f(x+h)-f(x))/(h)`

`f'(x)=lim_(x\rightarrow 0)((6+(x+h))/(1-6(x+h))-(6+x)/(1-6x))/(h)`

`f'(x)=lim_(x\rightarrow 0)(1)/(h)((6+x+h)(1-6x)-(6+x)(1-6x-6h))/((1-6x-6h)(1-6x))`

`f'(x)=lim_(x\rightarrow 0)(1)/(h)((6+x)(1-6x)+h(1-6x)-(6+x)(1-6x)-(6+x)(-6h))/((1-6x-6h)(1-6x))`

`f'(x)=lim_(x\rightarrow 0)(h)/(h)(1-6x+36+6x)/((1-6x-6h)(1-6x))`

`f'(x)=lim_(x\rightarrow 0)(37)/((1-6x-6h)(1-6x))`

Apply limit.

`f'(x)=(37)/((1-6x-6(0))(1-6x))`

`f'(x)=(37)/((1-6x)^2)`

The derivative of given function is
`f'(x)=(37)/((1-6x)^2)`.

`(1-6x)^2=0`

`1-6x=0`

`1=6x`

Divide both sides by 6.

`(1)/(6)=x`

The function f'(x) is not defined for 1/6. So, the domain of the function f'(x) is

`Domain=(-\infty,(1)/(6))\cup ((1)/(6),\infty)`.