I can see where the [tex] \frac{1}{2}\left[(t-1)^{-\frac{1}{2}}\right][/tex] came from but the [tex]t^{-3} \left(4-3t\right)[/tex] has lost me. What did you do to get that because I would have just done [tex]-2t^{-3}[/tex]dextercioby said:
A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is important because it helps us understand how a function changes over time, and is crucial in many areas of science, such as physics and economics.
To take the derivative of a function, you use a set of rules and formulas known as calculus. The specific method for taking the derivative depends on the type of function, but it generally involves finding the slope of the function at a specific point.
Yes, there are various shortcuts and techniques that can be used to simplify the process of taking a derivative. These include the power rule, product rule, quotient rule, and chain rule. It is important to understand these rules in order to efficiently take derivatives of more complex functions.
Derivatives have numerous real-world applications. For example, in physics, derivatives are used to calculate velocity and acceleration. In economics, derivatives are used to analyze and predict changes in supply and demand. In engineering, derivatives are used to optimize designs and improve efficiency.
Some common mistakes when taking derivatives include forgetting to apply the chain rule, mixing up signs or exponents, and not simplifying the final answer. It is important to double-check your work and use the appropriate rules and formulas to avoid these mistakes.