If [tex]f(x)=\frac{3}{x^6}[/tex], then, [tex]f^{\prime}(x)=\frac{[?]}{x}[/tex]
Answer (1)
Rewrite the function: f ( x ) = 3 x − 6 .
Apply the power rule: f ′ ( x ) = − 18 x − 7 .
Rewrite as a fraction: f ′ ( x ) = x 7 − 18 .
Express in the required form: f ′ ( x ) = x 8 − 18 x .
The derivative is x 8 − 18 x .
Explanation
Problem Setup We are given the function f ( x ) = f r a c 3 x 6 and we want to find its derivative, f ′ ( x ) , and express it in the form f r a c [ ?] x .
Rewrite the function First, rewrite f ( x ) using a negative exponent: f ( x ) = 3 x − 6
Apply the power rule Now, apply the power rule for differentiation, which states that if f ( x ) = a x n , then f ′ ( x ) = na x n − 1 . In our case, a = 3 and n = − 6 . Therefore, f ′ ( x ) = ( − 6 ) ( 3 ) x − 6 − 1 = − 18 x − 7
Rewrite as a fraction Rewrite f ′ ( x ) as a fraction: f ′ ( x ) = f r a c − 18 x 7
Manipulate the expression To express f ′ ( x ) in the form f r a c [ ?] x , we need to manipulate the expression. Multiply both the numerator and the denominator by x : f ′ ( x ) = f r a c − 18 x 7 = f r a c − 18 x x 7 c d o t x = f r a c − 18 x x 8
Final Answer Thus, f ′ ( x ) = f r a c − 18 x x 8 . The expression we are looking for is the numerator, which is − 18 x .
Examples
In physics, if f ( x ) represents the position of an object as a function of time x , then f ′ ( x ) represents the velocity of the object. For example, if the position of an object is given by f ( x ) = x 6 3 , then the velocity of the object at time x is given by f ′ ( x ) = x 7 − 18 . This is useful for understanding how the velocity changes over time.
