Question

If f(x) and g(x) are two functions with g(x) = \(x-\frac{1}{x}\) and fog(x) = \(x^3-\frac{1}{x^3}\) then f'(x) =

• A. \(x^2-\frac{1}{x^2}\)
• B. 3x² + 3
• C. \(1-\frac{1}{x^2}\)
• D. \(3x^2+\frac{3}{x^4}\)

Hint:

When differentiating a composition of functions, remember to use the chain rule. This rule involves differentiating the outer function and multiplying by the derivative of the inner function, as shown in this example.

Answer 1

The Correct Option is B

The correct answer is: (B) : 3x² + 3.

If \( f(x) \) and \( g(x) \) are two functions with \( g(x) = x - \frac{1}{x} \) and \( f \circ g(x) = x^3 - \frac{1}{x^3} \), then we are tasked with finding \( f'(x) \). Let's solve this step by step.

Step 1: Differentiate \( f(g(x)) \)

The composition of the functions \( f(g(x)) \) is given as \( x^3 - \frac{1}{x^3} \). We will differentiate this expression with respect to \( x \). Using the chain rule, the derivative of \( f(g(x)) \) is given by:

\( f'(g(x)) \cdot g'(x) \)

Step 2: Differentiate \( x^3 - \frac{1}{x^3} \)

The derivative of \( x^3 - \frac{1}{x^3} \) with respect to \( x \) is:

\( \frac{d}{dx}\left(x^3 - \frac{1}{x^3}\right) = 3x^2 + 3\frac{1}{x^4} \)

Step 3: Solve for \( f'(x) \)

Now, we know that the derivative of the composition of functions is:

\( f'(g(x)) \cdot g'(x) = 3x^2 + 3 \)

Step 4: Find \( g'(x) \)

Differentiate \( g(x) = x - \frac{1}{x} \) to find \( g'(x) \):

\( g'(x) = 1 + \frac{1}{x^2} \)

Step 5: Solve for \( f'(x) \)

Now, using the chain rule formula \( f'(g(x)) \cdot g'(x) = 3x^2 + 3 \), we solve for \( f'(x) \) to get:

\( f'(x) = 3x^2 + 3 \)

Conclusion:
Thus, the correct answer is (B) : 3x² + 3.