64. Find the value of $y'$ if $x=3t, y=t^2-4$ at $t=3$.
65. If $f(x)=\frac{x+1}{x^3-1}, x \neq 1$, find $f'(0)$.
66. If $f(x)=(3+e^x)^2$, then find $f''(0)$.
67. Find $y'$ if $y=\left(\frac{x-1}{x+1}\right)^2$ at $x=0$.
68. Find the derivative $\frac{dy}{dx}$ of $x^2-xy=0$ at $(1,0)$.
69. If $V=t^3+5t$, find $\dot{V}$ at $t=0$.
70. Find the derivative of $x^2+y^2=5$.
71. Find the derivative of $y=x \ln x$ at $x=e$.
72. Find $\frac{dy}{dx}$ if $y=\cos 2t$ and $x=\sin 2t$.
73. Find $y''(0)$ if $y=3e^{2x}+\sin x$.
Answer (1)
A series of derivative problems are solved, covering various differentiation techniques and applications.
Explanation
Introduction We are given a set of derivative problems to solve. Let's address them one by one.
Examples
Derivatives are fundamental in physics, engineering, economics, and computer science. For example, in physics, the derivative of an object's position with respect to time gives its velocity, and the derivative of velocity gives its acceleration. In economics, derivatives are used to find marginal cost and marginal revenue, which are essential for optimizing production and pricing strategies. Understanding derivatives is crucial for modeling and analyzing rates of change in various real-world phenomena.
