Find the derivative of $f(x)=2 x^3-5 x^2+3 x+2$ and place your answer below.
Answer: $f^{\prime}(x)=$ $\square$
Answer (1)
Apply the power rule to each term of the function f ( x ) = 2 x 3 − 5 x 2 + 3 x + 2 .
The derivative of 2 x 3 is 6 x 2 .
The derivative of − 5 x 2 is − 10 x .
The derivative of 3 x is 3 , and the derivative of 2 is 0 . Thus, f ′ ( x ) = 6 x 2 − 10 x + 3 .
Explanation
Problem Analysis We are given the function f ( x ) = 2 x 3 − 5 x 2 + 3 x + 2 and we need to find its derivative, f ′ ( x ) . To do this, we will apply the power rule to each term in the function. The power rule states that if we have a term of the form a x n , its derivative is na x n − 1 .
Applying the Power Rule Let's apply the power rule to each term:
For 2 x 3 , the derivative is 3 ⋅ 2 x 3 − 1 = 6 x 2 .
For − 5 x 2 , the derivative is 2 ⋅ ( − 5 ) x 2 − 1 = − 10 x .
For 3 x , the derivative is 1 ⋅ 3 x 1 − 1 = 3 .
For the constant term 2 , the derivative is 0 .
Combining the Derivatives Now, we combine the derivatives of each term to find the derivative of the entire function:
f ′ ( x ) = 6 x 2 − 10 x + 3
Final Answer Therefore, the derivative of the function f ( x ) = 2 x 3 − 5 x 2 + 3 x + 2 is f ′ ( x ) = 6 x 2 − 10 x + 3 .
Examples
In physics, if f ( x ) represents the position of an object at time x , then f ′ ( x ) represents the velocity of the object at time x . For example, if the position of a particle is given by f ( x ) = 2 x 3 − 5 x 2 + 3 x + 2 , then its velocity at any time x is given by f ′ ( x ) = 6 x 2 − 10 x + 3 . This allows us to determine how fast the particle is moving at any given moment.
