Question

Compute the derivative of the given function. $$g(t)=\frac{t^{5}-t^{3}}{e^{t}}$$

Short answer

The derivative is \( g'(t) = \frac{-t^5 + 5t^4 + t^3 - 3t^2}{e^t} \).

Step-by-step solution

Identify the Function Components

The function is given as \( g(t) = \frac{t^5 - t^3}{e^t} \). Here, the numerator is \( t^5 - t^3 \) and the denominator is \( e^t \).

Apply the Quotient Rule

Use the quotient rule \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \), where \( u = t^5 - t^3 \) and \( v = e^t \).

Differentiate the Numerator

Find the derivative of the numerator \( u = t^5 - t^3 \). Thus, \( u' = 5t^4 - 3t^2 \).

Differentiate the Denominator

Find the derivative of the denominator \( v = e^t \). Thus, \( v' = e^t \) since the derivative of \( e^t \) is \( e^t \) itself.

Compute the Derivative

Substitute the derivatives into the quotient rule: \( g'(t) = \frac{(5t^4 - 3t^2)e^t - (t^5 - t^3)e^t}{(e^t)^2} \).

Simplify the Expression

Simplify the expression by factoring out \( e^t \) in the numerator: \( g'(t) = \frac{e^t(5t^4 - 3t^2 - t^5 + t^3)}{e^{2t}} \).

Combine Like Terms

Combine like terms in the numerator: \( g'(t) = \frac{e^t(-t^5 + 5t^4 + t^3 - 3t^2)}{e^{2t}} \).

Cancel Out Terms

Cancel \( e^t \) in the numerator and one \( e^t \) in the denominator: \( g'(t) = \frac{-t^5 + 5t^4 + t^3 - 3t^2}{e^t} \).

Key concepts

Derivative

In mathematics, a derivative represents the rate at which a function is changing at any given point. It's a fundamental concept in calculus used to find the slope of the tangent line to a curve at a particular point. When you compute the derivative of a function, you essentially measure how the function's value changes as its input changes.Understanding derivatives is essential because:

• They help in finding maxima and minima of functions which is useful in optimization problems.
• They can be used to model rates of change in various fields such as physics, economics, and biology.

To find the derivative of a quotient like in our problem, we use the Quotient Rule. It applies specifically to functions that are the ratio of two other functions. If you have a function expressed as \( \ \frac{u}{v} \) , the derivative is computed as:\[\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}\]This rule requires finding the derivative of both the numerator \( u \) and the denominator \( v \) separately before using them in the formula. It’s crucial to practice this rule as it applies to many real-world problems.

Exponential Function

An exponential function is a type of mathematical function in the form \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. The function \( e^t \) plays a significant role in various scientific fields due to its unique properties such as its rate of growth being proportional to its current value.Key aspects of exponential functions include:

• The base \( e \), which is an irrational and transcendental number.
• Its derivative \( \frac{d}{dt} e^t = e^t \), meaning the slope of the tangent line at any point on the curve is equal to the value of the function at that point.
• Applications range from modeling populations, radioactive decay, to interest rates in finance.

In our problem, the denominator of the function is \( e^t \). When finding its derivative, it remains \( e^t \), emphasizing that the exponential function grows at the same rate as its value changes.

Polynomial Function

Polynomial functions comprise terms made up of variables raised to whole-number exponents and their corresponding coefficients. A typical example of a polynomial function is \( t^5 - t^3 \), which places these terms in descending order by their degree.Characteristics of polynomial functions include:

• Their smooth, continuous curves without breaks or sharp angles.
• One can easily find the derivative using basic power rules where \( \frac{d}{dt}t^n = nt^{n-1} \).
• Polynomials are widely used in modeling and can approximate other functions under certain conditions.

For the polynomial \( t^5 - t^3 \), applying the power rule gives the derivative as \( 5t^4 - 3t^2 \). Calculating derivatives for polynomial functions is generally straightforward, making them a favorite starting point for those new to calculus. Understanding these derivatives aids greatly in comprehending how polynomial functions change and form the basis for dealing with more complex functions.