If [tex]$y=3 t^3+1$[/tex], then find [tex]$\frac{d y}{d t}$[/tex]
Answer (2)
The derivative of the function y = 3 t 3 + 1 with respect to t is d t d y = 9 t 2 . This is calculated using the power rule of differentiation. The constant term does not contribute to the derivative as its derivative is zero.
;
The problem asks to find the derivative of y = 3 t 3 + 1 with respect to t .
Apply the power rule to differentiate 3 t 3 , which gives 9 t 2 .
The derivative of the constant 1 is 0 .
Thus, the derivative d t d y is 9 t 2 .
Explanation
Problem Analysis We are given the function y = 3 t 3 + 1 and asked to find its derivative with respect to t , which is denoted as d t d y . This involves applying the basic rules of differentiation.
Differentiation Rules To find d t d y , we will differentiate y = 3 t 3 + 1 with respect to t . We apply the power rule, which states that d x d ( x n ) = n x n − 1 , and the constant rule, which states that the derivative of a constant is 0. Also, we'll use the constant multiple rule which states that d x d ( c f ( x )) = c d x d f ( x ) .
Applying Differentiation Applying these rules, we have: d t d y = d t d ( 3 t 3 + 1 ) d t d y = d t d ( 3 t 3 ) + d t d ( 1 ) d t d y = 3 d t d ( t 3 ) + 0 d t d y = 3 ( 3 t 2 ) d t d y = 9 t 2
Final Result Therefore, the derivative of y with respect to t is 9 t 2 .
Examples
In physics, if y represents the position of an object at time t , then d t d y represents the object's velocity. For example, if the position of a particle is given by y = 3 t 3 + 1 , then its velocity at time t is 9 t 2 . This tells us how fast the particle is moving at any given time. Understanding derivatives is crucial for analyzing motion and other dynamic processes.
