If [tex]p(u)=\int_0^{u^3} \sqrt{8+r^4} dr[/tex] then [tex]p^{\prime}(u)=[/tex] ?
Answer (2)
The derivative of the function p ( u ) defined by the integral p ( u ) = ∫ 0 u 3 8 + r 4 d r is p ′ ( u ) = 3 u 2 8 + u 12 . This is derived using the Fundamental Theorem of Calculus and the chain rule. Each step involves substitution and differentiation, resulting in the final expression for p ′ ( u ) .
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Apply the Fundamental Theorem of Calculus to recognize that the derivative of the integral will involve the original function evaluated at the upper limit of integration.
Use the chain rule because the upper limit of integration is a function of u .
Substitute and simplify the expression to find the derivative.
The derivative p ′ ( u ) is 3 u 2 8 + u 12 .
Explanation
Problem Analysis We are given the function $p(u) =
\int_0^{u^3} \sqrt{8+r^4} dr an d w e w an tt o f in d i t s d er i v a t i v e p'(u)$. This problem involves the Fundamental Theorem of Calculus and the chain rule.
Applying the Fundamental Theorem of Calculus Let F ( r ) be the antiderivative of 8 + r 4 , so F ′ ( r ) = 8 + r 4 . Then p ( u ) = F ( u 3 ) − F ( 0 ) .
Differentiating with Chain Rule Now, we differentiate p ( u ) with respect to u using the chain rule:
p ′ ( u ) = d u d [ F ( u 3 ) − F ( 0 )] = F ′ ( u 3 ) ⋅ d u d ( u 3 ) − 0
Substituting the Derivative Since F ′ ( r ) = 8 + r 4 , we have F ′ ( u 3 ) = 8 + ( u 3 ) 4 . Also, d u d ( u 3 ) = 3 u 2 .
Simplifying the Expression Therefore, p ′ ( u ) = 8 + ( u 3 ) 4 ⋅ 3 u 2 = 3 u 2 8 + u 12 .
Final Answer Thus, the derivative of p ( u ) with respect to u is p ′ ( u ) = 3 u 2 8 + u 12 .
Examples
In physics, if p ( u ) represents the amount of energy required to accelerate a particle to a velocity related to u , then p ′ ( u ) gives the rate of change of this energy with respect to u . This can be crucial in understanding the power requirements of particle accelerators or in analyzing the energy dynamics of high-speed collisions. The derivative helps physicists optimize energy usage and predict particle behavior under various conditions, making it a vital tool in advanced physics research and engineering.
