Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=-\frac{3}{(t-2)^{4}} $$

Short answer

The derivative of the function \(y=-\frac{3}{(t-2)^{4}}\) is \(y' = \frac{12}{(t-2)^5}\)

Step-by-step solution

Identifying the outer and inner functions

Looking at the function \(y=-\frac{3}{(t-2)^{4}}\), it can be decomposed into an outer function \(y=-3u\) and an inner function \(u=(t-2)^{-4}\). Where \(u\) is chosen to be a function of \(t\). This is a prerequisite to apply the chain rule.

Computing Derivatives of Inner and Outer Function

The derivative of the outer function \(y'=-3\) and the derivative of the inner function u is found by applying power rule. The power rule states that the derivative of \(u=t^n\) is \(u'=nt^{n-1}\). Hence the derivative of \(u\) is \(u'=(-4)(t-2)^{-5}\).

Applying Chain Rule of Derivatives

The chain rule of derivatives states that the derivative of a composite function is the derivative of the outer function evaluated at the inner, multiplied by the derivative of the inner. The derivative of the function \(y\) is therefore \(y' = y'u'u' = -3(-4)(t-2)^{-5}\).

Simplifying Expression for Final Answer

Multiplying, \(y' = 12(t-2)^{-5}\). Solving for -5 as exponent to bring it to the denominator. We get \(y' = \frac{12}{(t-2)^5}\).

Key concepts

Chain Rule

Understanding the chain rule is crucial for anyone diving into the world of calculus. It is a method used to find the derivative of composite functions, or functions within functions. Imagine you have a Russian doll, where one is nested within the other – that's the idea behind composite functions. The chain rule allows you to open each doll sequentially to reveal the one inside.

When using the chain rule, you differentiate the outer

Power Rule

The power rule is a swift shortcut in calculus, used to find the derivative of functions where the variable has an exponent. Essentially, it's like a quick leap in a parkour run – it gets you where you need to go faster and more efficiently. The power rule states that if you have a function,

Composite Function

A composite function is a function that combines one function inside another. Think of it as a recipe that calls for a smaller recipe within it. You must complete the inner recipe first before integrating it into the larger one. In calculus, a composite function is usually written as

Calculus

Calculus is the branch of mathematics that deals with change and motion. It's the toolbox for understanding the dynamics of the physical world, from the orbits of planets to the growth of populations. This mathematical frontier is divided into two sub-fields: differential calculus, focusing on the concept of the derivative, and integral calculus, which looks at the antiderivative or the integral.

The derivative, which we've seen practiced in our problem set, is a measure of how a function changes as its input changes. It's like the speedometer in a car that tells you how quickly you're going at any given moment. Calculus is about much more than just solving problems – it's a way of seeing the world, a lens that reveals the hidden patterns behind the complex behaviors we observe in nature.