Questions: What is the derivative of f(x)=x^n where n is a constant?

Solution Steps

The function given is \( f(x) = x^n \), where \( n \) is a constant. To find its derivative, we will use the power rule for differentiation. The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) \) is:

\[ f'(x) = n x^{n-1}. \]

Using the power rule, we differentiate \( f(x) = x^n \) with respect to \( x \):

\[ f'(x) = n x^{n-1}. \]

Now, we compare the result \( f'(x) = n x^{n-1} \) with the provided options:

• a. \( f'(x) = n x^{n-1} \)
• b. \( f'(x) = n x \)
• c. \( f'(x) = x^{n-1} \)
• d. \( f'(x) = x^{n+1} \)

The correct answer matches option a.

Final Answer