If [tex]$y=3 t^{\prime 3}+[/tex], then find [tex]$\frac{d y}{d t}$[/tex]
Answer (2)
The derivative of the function y = 3 t 3 + c with respect to t is d t d y = 9 t 2 . This is derived by applying the power rule of differentiation and noting that the derivative of a constant is zero.
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Identify the function: y = 3 t 3 + c .
Apply the power rule to the t 3 term: d t d ( 3 t 3 ) = 9 t 2 .
Recognize that the derivative of a constant is zero.
State the final derivative: d t d y = 9 t 2 .
Explanation
Problem Setup We are given the function y = 3 t 3 + c , where c is a constant, and we want to find its derivative with respect to t , which is denoted as d t d y .
Applying the Power Rule To find the derivative, we will use the power rule. The power rule states that if y = a x n , then d x d y = na x n − 1 . In our case, we have y = 3 t 3 + c .
Differentiating the Term Applying the power rule to the term 3 t 3 , we get d t d ( 3 t 3 ) = 3 ⋅ 3 t 3 − 1 = 9 t 2 .
Differentiating the Constant The derivative of a constant is zero. Therefore, the derivative of the constant term c is 0.
Final Derivative Combining the results, we find that d t d y = 9 t 2 + 0 = 9 t 2 .
Examples
In physics, if y represents the position of an object at time t , then d t d y represents the velocity of the object. For example, if the position of an object is given by y = 3 t 3 + 5 , then the velocity of the object at time t is d t d y = 9 t 2 . This tells us how the object's position changes with respect to time.
