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The quotient rule is a method used in calculus to find the derivative of a function that is the ratio of two differentiable functions. It is specifically useful when you have a fraction where both the numerator and the denominator are functions of the same variable.
Here’s the formula for the quotient rule:
\[ \frac{d}{dx} \frac{u}{v} = \frac{u'v - uv'}{v^2} \]
In this formula, \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives. To apply this rule:

• Differentiate the numerator (\( u \)) to get \( u' \)
• Differentiate the denominator (\( v \)) to get \( v' \)
• Plug these into the quotient rule formula
• Simplify the expression if possible

For example, consider the original problem:
\[ \frac{t}{5 + 2t} \]
Here, \( u = t \) and \( v = 5 + 2t \). After differentiating:\(
u' = 1 \) and \( v' = 2 \).
Now, substitute these into the quotient rule formula:
\[ \frac{1(5 + 2t) - t(2)}{(5 + 2t)^2} = \frac{5}{(5 + 2t)^2} \]
This gives you the derivative of the first term.

The power rule is a basic rule used to find the derivative of polynomial functions. It states that:
\[ \frac{d}{dx} t^n = nt^{n-1} \]
Here, \( n \) is any real number. This rule helps in quickly finding the derivatives of terms where a variable is raised to a power.

• The exponent \( n \) comes down as a coefficient
• The new exponent becomes \( n-1 \)

Let's apply this rule to a term from the example:
\[ -2t^4 \]
Using the power rule: \[ \frac{d}{dt} (-2t^4) = -2 \times 4t^{4-1} = -8t^3 \]
The power rule simplifies the process of differentiation for polynomial terms, providing a straightforward way to find the slope of the curve at any given point.

A polynomial function is a mathematical expression comprising variables, coefficients, and exponents that are non-negative integers. Differentiating polynomial functions involves breaking them down into their individual terms and then applying the power rule to each term.

• Identify each term in the polynomial
• Apply the power rule to each term
• Combine the results to get the complete derivative

Let's review the polynomial in the given problem:
\[ f(t)=\frac{t}{5+2t}-2t^4 \]
After separating the terms, we have:
\[ \frac{t}{5+2t} - 2t^4 \]
We've already differentiated these terms separately by using the quotient rule for \( \frac{t}{5+2t} \), and the power rule for \( -2t^4 \). Combine them to get the final derivative:
\[ f'(t) = \frac{5}{(5+2t)^2} - 8t^3 \]
By understanding polynomial differentiation, you can tackle a wide range of functions with various powers and coefficients.