If \( f(x)=x^{m+4} \), find \( f^{\prime}(x) \) at point \( (1,2) \)

To find \( f^{\prime}(x) \) for the function \( f(x) = x^{m+4} \), we will first differentiate the function with respect to \( x \). 1. **Differentiate the function:** The derivative of \( f(x) = x^{m+4} \) can be found using the power rule, which states that if \( f(x) = x^n \), then \( f^{\prime}(x) = n \cdot x^{n-1} \). Applying the power rule: \[ f^{\prime}(x) = (m+4) \cdot x^{(m+4)-1} = (m+4) \cdot x^{m+3} \] 2. **Evaluate the derivative at the point \( x = 1 \):** Now we need to find \( f^{\prime}(1) \): \[ f^{\prime}(1) = (m+4) \cdot 1^{m+3} = (m+4) \cdot 1 = m + 4 \] 3. **Given the point \( (1, 2) \):** The point \( (1, 2) \) indicates that when \( x = 1 \), \( f(1) = 2 \). We can use this information to find \( m \): \[ f(1) = 1^{m+4} = 1 = 2 \] This implies that \( 1^{m+4} \) does not equal 2, which means we need to find \( m \) such that \( f(1) = 2 \). Since \( 1^{m+4} = 1 \) for any \( m \), we cannot find \( m \) from this equation. However, we can conclude that the derivative \( f^{\prime}(1) \) is independent of the value of \( f(1) \). Thus, the final answer for \( f^{\prime}(1) \) is: \[ f^{\prime}(1) = m + 4 \]