Find the derivative of: [tex]$-7 x^3-4 x^8$[/tex]
Answer (1)
Apply the power rule to the first term: d x d ( − 7 x 3 ) = − 21 x 2 .
Apply the power rule to the second term: d x d ( − 4 x 8 ) = − 32 x 7 .
Combine the derivatives of each term.
The derivative of the function is: − 21 x 2 − 32 x 7 .
Explanation
Problem Analysis We are given the function f ( x ) = − 7 x 3 − 4 x 8 and we need to find its derivative.
Applying the Power Rule To find the derivative, we will use the power rule, which states that if f ( x ) = a x n , then f ′ ( x ) = na x n − 1 .
Derivative of the First Term First, let's find the derivative of the term − 7 x 3 . Using the power rule, we have: d x d ( − 7 x 3 ) = − 7 ⋅ 3 x 3 − 1 = − 21 x 2
Derivative of the Second Term Next, let's find the derivative of the term − 4 x 8 . Using the power rule, we have: d x d ( − 4 x 8 ) = − 4 ⋅ 8 x 8 − 1 = − 32 x 7
Combining the Derivatives Now, we combine the derivatives of each term to find the derivative of the entire function: f ′ ( x ) = − 21 x 2 − 32 x 7
Final Answer Therefore, the derivative of the function f ( x ) = − 7 x 3 − 4 x 8 is: f ′ ( x ) = − 21 x 2 − 32 x 7
Examples
In physics, if x ( t ) = − 7 t 3 − 4 t 8 represents the position of an object at time t , then the derivative x ′ ( t ) = − 21 t 2 − 32 t 7 gives the object's velocity at time t . Understanding derivatives allows us to analyze how the object's velocity changes over time, which is crucial in many physics and engineering applications.
