Question

Calculate the derivative of the following functions. $$y=\left((x+2)\left(x^{2}+1\right)\right)^{4}$$

Short answer

Question: Find the derivative of the function $$y = \left((x+2)(x^2 + 1)\right)^4$$ with respect to x. Answer: The derivative of the given function is $$\frac{dy}{dx} = 4\left((x+2)(x^2 + 1)\right)^3 \cdot (2x(x+2) + x^2 + 1)$$

Step-by-step solution

Identify the function's components

We have: $$y = \left((x+2)(x^2 + 1)\right)^4$$ This is a composite function, we can break it down into its components: $$u(x) = (x+2)(x^2 + 1)$$ $$y = u(x)^4$$

Find the derivative of u(x)

We have: $$u(x) = (x+2)(x^2 + 1)$$ Now we will use the product rule to find the derivative of u(x): $$\frac{du}{dx} = (x+2)\frac{d(x^2 + 1)}{dx} + (x^2 + 1)\frac{d(x+2)}{dx}$$ $$\frac{du}{dx} = (x+2)(2x) + (x^2 + 1)(1)$$ $$\frac{du}{dx} = 2x(x+2) + x^2 + 1$$

Find the derivative of y(u)

We have: $$y = u^4$$ Now, find the derivative of y with respect to u: $$\frac{dy}{du} = 4u^3$$

Apply the Chain Rule

To find the derivative of y with respect to x, we apply the chain rule: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Substitute and Simplify

Now we substitute the expressions of \(\frac{dy}{du}\) and \(\frac{du}{dx}\) using the results from step 2 and step 3: $$\frac{dy}{dx} = 4u^3 \cdot (2x(x+2) + x^2 + 1)$$ Now, replace u(x) with its original expression: $$\frac{dy}{dx} = 4\left((x+2)(x^2 + 1)\right)^3 \cdot (2x(x+2) + x^2 + 1)$$ The derivative of the given function y is: $$\frac{dy}{dx} = 4\left((x+2)(x^2 + 1)\right)^3 \cdot (2x(x+2) + x^2 + 1)$$

Key concepts

Chain Rule

Understanding the chain rule is essential for differentiating composite functions. Imagine you are wearing a glove and beneath that, a hand — the glove is a function of your hand, and your hand, in turn, is a function of its position. Similarly, the chain rule deals with functions nested within each other. In mathematical terms, if we have two functions, one inside another, the chain rule tells us how to find the derivative of this composite structure.

Applying the chain rule involves multiple steps:

• First, identify the outer function and differentiate it with respect to the inner function.
• Next, differentiate the inner function with respect to the independent variable.
• Finally, multiply these derivatives together.

The derivative of the entire composite function is the product of these two derivatives, symbolized as \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\) when \(u\) represents the inner function and \(y\) the outer function. Thus, it is akin to 'unstacking' the functions, differentiating each layer separately, and then stacking the derivatives back together.

Product Rule

When it comes to differentiating functions that are multiplied together, we call upon the product rule — a fundamental technique in calculus. Similar to how you would attentively dismantle a stack of building blocks instead of just knocking it down, the product rule allows us to differentiate products of functions meticulously.

Here's a brief explanation of this technique:

• Given two functions \(u(x)\) and \(v(x)\), we don't simply differentiate each and multiply the results.
• Instead, we apply the product rule: \(\frac{d}{dx}[u(x) \cdot v(x)] = u(x) \frac{dv}{dx} + v(x) \frac{du}{dx}\).

This says that the derivative of a product is the first function times the derivative of the second, plus the second function times the derivative of the first. It is based on the distributive property in algebra and ensures that no part of our function's rate of change is overlooked.

Composite Function

In calculus, a composite function can be visualized like a Russian nesting doll, where one function is nested inside another. Formally, if you have two functions \(f\) and \(g\), then the composite function \(f(g(x))\) means that \(g\) is applied first and \(f\) is applied to the result.

Understanding composite functions is crucial since many real-world scenarios can be modeled this way, such as applying tax after a discount on a price. When you calculate the derivative of a composite function, you are essentially finding out how the entire 'nest' of functions changes as the innermost variable changes. This concept is where the majority of differentiation rules, including the chain rule, come into play. It's a core concept that makes complex relationships and changes more manageable by breaking them down into simpler parts.