If [tex]f(x)=7 x+10[/tex], find [tex]f^{\prime}(x)[/tex].
Answer (1)
Recognize that the derivative of a linear function f ( x ) = a x + b is simply its slope.
Apply the power rule to find the derivative of 7 x , which is 7 .
Note that the derivative of the constant 10 is 0 .
Conclude that f ′ ( x ) = 7 .
Explanation
Problem Analysis We are given the function f ( x ) = 7 x + 10 and asked to find its derivative, f ′ ( x ) .
Differentiation Rules To find the derivative, we will apply 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 The derivative of 7 x is 7 ⋅ 1 ⋅ x 1 − 1 = 7 ⋅ 1 ⋅ x 0 = 7 ⋅ 1 ⋅ 1 = 7 . The derivative of the constant 10 is 0 .
Final Calculation Therefore, 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 + 10 , where x is time in seconds and f ( x ) is the position in meters, then the velocity of the object is f ′ ( x ) = 7 meters per second. This means the object is moving at a constant velocity of 7 m/s.
