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The first derivative of a function helps you understand how the function is changing at any point. It provides the slope, or rate of change, of the function. By applying the first derivative, you can discover where the function is increasing, decreasing, or staying constant. In our given function, \(g(t)=-t^{3}+12 t^{2}+36 t+45\), finding the first derivative involves using the power rule extensively. Recall that the power rule for differentiation is expressed as \(\frac{d}{dt}[t^n] = nt^{n-1}\). Using this, each term in the function is differentiated:- The derivation of \(-t^3\) becomes \(-3t^2\).- \(12t^2\) turns into \(24t\).- The term \(36t\) becomes \(36\), since \(t^1\) simplifies to 1.Thus, putting these together gives us the first derivative: \(g'(t) = -3t^2 + 24t + 36\). Now, we know the rate at which our function changes for any particular value of \(t\). By analyzing this expression, one can determine critical points and behavior of the graph at different intervals.

The second derivative provides insight into the concavity of the function. It tells us whether the function is curving upwards or downwards. More formally, the second derivative reveals the acceleration of the change described by the first derivative.To find the second derivative, differentiate the first derivative \(g'(t) = -3t^2 + 24t + 36\) again. Leveraging the power rule again:- Differentiating \(-3t^2\) results in \(-6t\).- The derivation of \(24t\) yields \(24\).- Constant terms like \(36\) disappear when differentiated since they do not change with \(t\).Combining these, the second derivative is \(g''(t) = -6t + 24\).The second derivative is key in identifying potential inflection points, where the concavity of the function changes. Understanding this concept helps you anticipate the behavior and the overall shape of the function's graph.

The power rule is a fundamental tool in calculus for finding derivatives of polynomial expressions. It simplifies the process of differentiation by providing a straightforward mechanism to compute the derivative of each term in the function.Here's how the power rule works:- For a term \(t^n\), apply the rule \(\frac{d}{dt}[t^n] = nt^{n-1}\).When faced with a polynomial, like our function \(g(t)=-t^{3}+12 t^{2}+36 t+45\), apply the power rule term by term:- For \(-t^3\), multiply the exponent by the coefficient and subtract one from the exponent: \(-3t^{2}\).- In \(12t^2\), follow the same by getting \(24t\).- \(36t\) leads to \(36\) by applying the rule.The rule provides a quick shortcut, making it easy to handle otherwise complex differentiation processes. It's essential for solving a wide range of problems involving rates of change, slopes of curves, and motion prediction over time. The power rule connects seamlessly with other derivative concepts, such as the first and second derivatives, enhancing our comprehension of a function’s character and behavior.