Derivatives Find and simplify the derivative of the following functions.
g(t) = t³+3t²+t / t³

검증된 단계별 안내

Step 1: Simplify the function g(t) = \(\frac{t^3 + 3t^2 + t}{t^3}\) by dividing each term in the numerator by t^3. This gives g(t) = 1 + \(\frac{3}{t}\) + \(\frac{1}{t^2}\).

Step 2: Rewrite the function in terms of powers of t: g(t) = 1 + 3t^{-1} + t^{-2}.

Step 3: Differentiate each term separately using the power rule. The power rule states that the derivative of t^n is n*t^{n-1}.

Step 4: Apply the power rule: The derivative of 1 is 0, the derivative of 3t^{-1} is -3t^{-2}, and the derivative of t^{-2} is -2t^{-3}.

Step 5: Combine the derivatives to find g'(t) = 0 - 3t^{-2} - 2t^{-3}. Simplify to get g'(t) = -\(\frac{3}{t^2}\) - \(\frac{2}{t^3}\).

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주요 개념

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Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function in the form f(t) = u(t)/v(t), the derivative is given by f'(t) = (u'v - uv')/v², where u and v are differentiable functions of t. This rule is essential when dealing with functions that are divided by another function.

Simplification of Derivatives

After finding the derivative of a function, simplification is often necessary to express the result in its simplest form. This may involve factoring, reducing fractions, or combining like terms. Simplifying the derivative can make it easier to analyze the function's behavior, such as identifying critical points or determining concavity.

Derivatives Find and simplify the derivative of the following functions.g(t) = t³+3t²+t / t³