Given $f(x)=5 x+5$, find $f^{\prime}(x)$ using the limit definition of the derivative.
$f^{\prime}(x) =$
Answer (1)
Find f ( x + h ) : Substitute x + h into f ( x ) to get f ( x + h ) = 5 ( x + h ) + 5 = 5 x + 5 h + 5 .
Apply the limit definition: Substitute f ( x + h ) and f ( x ) into the formula f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x ) .
Simplify the expression: Simplify the fraction inside the limit to h 5 h .
Evaluate the limit: Cancel h and find the limit as h approaches 0, resulting in 5 .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 x + 5 and asked to find its derivative, f ′ ( x ) , using the limit definition of the derivative. This means we need to use the formula: f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) .
Finding f(x+h) First, we need to find f ( x + h ) . We substitute x + h into the function f ( x ) : f ( x + h ) = 5 ( x + h ) + 5 = 5 x + 5 h + 5 .
Substituting into Limit Definition Now, we substitute f ( x + h ) and f ( x ) into the limit definition: f ′ ( x ) = h → 0 lim h ( 5 x + 5 h + 5 ) − ( 5 x + 5 ) .
Simplifying the Expression Next, we simplify the expression inside the limit: f ′ ( x ) = h → 0 lim h 5 x + 5 h + 5 − 5 x − 5 = h → 0 lim h 5 h .
Canceling h Now, we cancel h from the numerator and the denominator: f ′ ( x ) = h → 0 lim 5 .
Evaluating the Limit Finally, we evaluate the limit. Since there is no h in the expression, the limit is simply 5: f ′ ( x ) = 5 .Therefore, the derivative of f ( x ) = 5 x + 5 is 5.
Final Answer The derivative of the function f ( x ) = 5 x + 5 using the limit definition is 5 .
Examples
In physics, if f ( x ) represents the position of an object at time x , then f ′ ( x ) represents the object's velocity. For example, if the position of an object is given by f ( x ) = 5 x + 5 , where x is time in seconds and f ( x ) is the position in meters, then the velocity of the object is f ′ ( x ) = 5 meters per second. This means the object is moving at a constant velocity of 5 m/s.
