Question

Let f be the function defined by f(x)=bx²+3bx+b²+ 1/x² , where b is a nonzero constant. Which of the following is an expression for f'(x) , the derivative of f(x) ?

Answer 1

To find the derivative of the function f(x), we use the power rule and the product rule. The expression for f'(x) is 2bx + 3b - 2/x³.

Step-by-step explanation:

The function f(x) is defined as f(x) = bx² + 3bx + b² + 1/x². To find the derivative f'(x), we need to use the power rule and the product rule. First, let's differentiate each term separately:

• For the term bx², the derivative is 2bx.
• For the term 3bx, the derivative is 3b.
• For the term b², the derivative is 0 since b² is a constant.
• For the term 1/x², we can rewrite it as x^(-2) and apply the power rule. The derivative is (-2)x^(-3) = -2/x^3.

Now, add up all the derivatives: f'(x) = 2bx + 3b - 2/x^3. Therefore, the expression for f'(x) is 2bx + 3b - 2/x³.