Question

Consider the following functions.

g(x) = x + √7x
h(x) = 7x − 4
f(x) = x + √7x / 7x − 4
Find the derivative of each function.
g '(x)=
h '(x)=
f '(x)=

Answer 1

To find the derivatives of the given functions, we will use the rules of differentiation. The derivative of g(x) is 1 + (1/2)√7, the derivative of h(x) is 7, and the derivative of f(x) is given by a specific expression involving the previous derivatives we found.

Step-by-step explanation:

To find the derivative of each function, we will use the rules of differentiation. Let's start with g(x) = x + √7x. Using the power rule and the sum rule, we can find g'(x) as follows:

1. Derivative of x with respect to x is 1.
2. Derivative of √7x with respect to x is (1/2)√7.
3. Add the derivatives: g'(x) = 1 + (1/2)√7.

Now, let's move on to h(x) = 7x - 4. Since h(x) is a linear function, the derivative is simply the coefficient of x, which is 7. Therefore, h'(x) = 7.

Finally, let's find the derivative of f(x) = (x + √7x) / (7x - 4). Using the quotient rule and the previous derivatives we found, we have:

1. Derivative of x + √7x is 1 + (1/2)√7.
2. Derivative of 7x - 4 is 7.
3. Apply the quotient rule: f'(x) = [(7)(1 + (1/2)√7) - (1 + (1/2)√7)(7)] / [(7x - 4)^2].

Therefore, the derivatives of the given functions are: g'(x) = 1 + (1/2)√7, h'(x) = 7, and f'(x) = [(7)(1 + (1/2)√7) - (1 + (1/2)√7)(7)] / [(7x - 4)^2].