Question

I really need help with this problem! it’s calculus. i need to find the derivative and domain of a system of equation.

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Define the domain of the given piecewise function

Given the conditions for the piecewise function, it can be seen that the function starts from negative infinity to positive infinity. Therefore, the domain of the given piecewise function is all set of Real numbers.

STEP 2: Differentitate the given piecewise functions

For the first function,

`\begin{gathered} f(x)=5x-6 \\ derivative=(d(5x))/(dx)-(d)/(dx)(6)=5-0=5 \end{gathered}`

For the second function,

`\begin{gathered} f(x)=x^2 \\ (d)/(dx)(x^2)=2x \end{gathered}`

For the third function,

`\begin{gathered} f(x)=2x+8 \\ (d)/(dx)(2x)+(d)/(dx)(8)=2+0=2 \end{gathered}`

STEP 3: Determie the domain of the differentiated functions

When you differentitate a piecewise function, the domain at the given points will be undefined. Therefore, these points will be excluded. These points are x=2 and x=4. The domain of a piecewise function after being differentitated will be;

`All\text{ }real\text{ }numbers,x\\e2,x\\e4`

Hence, the answer is given as: