If [tex]$y=4 x^3+2 x+1$[/tex], then find [tex]$\frac{d y}{d x}$[/tex]
Answer (2)
The derivative of the function y = 4 x 3 + 2 x + 1 is d x d y = 12 x 2 + 2 . This is found by applying the power rule to each term in the function. Each term is differentiated step-by-step, resulting in the final answer.
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Apply the power rule to the term 4 x 3 : d x d ( 4 x 3 ) = 12 x 2 .
Apply the power rule to the term 2 x : d x d ( 2 x ) = 2 .
Differentiate the constant term 1 : d x d ( 1 ) = 0 .
Sum the derivatives to find the final answer: 12 x 2 + 2 .
Explanation
Problem Analysis We are given the function y = 4 x 3 + 2 x + 1 and asked to find its derivative with respect to x , denoted as d x d y . This involves applying the power rule to each term in the function.
Power Rule The power rule states that if y = a x n , then d x d y = na x n − 1 . We will apply this rule to each term in the given function.
Differentiating the First Term Let's differentiate the first term, 4 x 3 . Using the power rule, we have: d x d ( 4 x 3 ) = 3 ⋅ 4 x 3 − 1 = 12 x 2
Differentiating the Second Term Now, let's differentiate the second term, 2 x . Using the power rule, we have: d x d ( 2 x ) = 1 ⋅ 2 x 1 − 1 = 2 x 0 = 2
Differentiating the Constant Term Next, we differentiate the constant term, 1 . The derivative of a constant is always 0: d x d ( 1 ) = 0
Summing the Derivatives Finally, we sum the derivatives of each term to find the derivative of y with respect to x :
d x d y = 12 x 2 + 2 + 0 = 12 x 2 + 2
Final Answer Therefore, the derivative of y = 4 x 3 + 2 x + 1 with respect to x is d x d y = 12 x 2 + 2 .
Examples
In physics, if y represents the position of an object at time x , then d x d y represents the object's velocity. For example, if the position of a particle is given by y = 4 x 3 + 2 x + 1 , then its velocity at any time x is given by d x d y = 12 x 2 + 2 . This tells us how the particle's position changes as time progresses, which is crucial for understanding its motion.
