If $f(x)=7 x+5$, find $f^{\prime}(x)$
Answer (1)
Identify the function: f ( x ) = 7 x + 5 .
Apply the power rule to the term 7 x , resulting in 7 .
Recognize that the derivative of the constant 5 is 0 .
Combine the results to find the derivative: 7 .
Explanation
Problem Analysis We are given the function f ( x ) = 7 x + 5 , and we want to find its derivative, f ′ ( x ) .
Differentiation Rules To find the derivative, we will use the power rule and the constant multiple rule. The power rule states that if f ( x ) = a x n , then f ′ ( x ) = na x n − 1 . The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function.
Applying the Rules Applying these rules to our function f ( x ) = 7 x + 5 , we can rewrite it as f ( x ) = 7 x 1 + 5 . The derivative of 7 x 1 is 1 × 7 x 1 − 1 = 7 x 0 = 7 × 1 = 7 . The derivative of the constant 5 is 0 .
Final Derivative Therefore, the derivative of f ( x ) = 7 x + 5 is f ′ ( x ) = 7 + 0 = 7 .
Examples
In physics, if f ( x ) represents the position of an object at time x , then f ′ ( x ) represents the velocity of the object. For example, if the position of an object is given by f ( x ) = 7 x + 5 , where x is time in seconds and f ( x ) is the position in meters, then the velocity of the object is constant and equal to 7 m/s. This means the object is moving at a constant rate in a straight line.
