Question

Use the Chain Rule combined with other differentiation rules to find 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 \(y = ((x+2)(x^2+1))^4\) using the Chain Rule combined with other differentiation rules. Answer: The derivative of the given function is \(\frac{dy}{dx} = 4((x+2)(x^{2}+1))^{3}(4x^2 + 4x + 2)\).

Step-by-step solution

Identify the Outer and Inner Functions

The given function is $$y=\left((x+2)\left(x^{2}+1\right)\right)^{4}$$ The outer function is \(u^4\), where \(u=(x+2)(x^2+1)\), and the inner function is \(u=(x+2)(x^2+1)\).

Differentiate the Outer Function

Using the Chain Rule, we will differeniate the outer function with respect to the inner function. The derivative of \(u^4\) with respect to \(u\) is $$\frac{dy}{du} = \frac{d}{du}(u^{4}) = 4u^{3}$$.

Differentiate the Inner Function

Next, we will differentiate the inner function \(u=(x+2)(x^2+1)\) with respect to \(x\). Using the Product Rule for differentiation, the derivative is $$\frac{du}{dx}=\frac{d}{dx}((x+2)(x^{2}+1)) = (x^{2}+1) \cdot \frac{d}{dx}(x+2) + (x+2) \cdot \frac{d}{dx}(x^{2}+1)$$. Now, differentiate each of the individual terms, $$\frac{d}{dx}(x+2) = 1, \, \frac{d}{dx}(x^{2}+1) = 2x$$. Substituting these values into the previous equation, we get $$\frac{du}{dx}=(x^{2}+1) \cdot 1 + (x+2) \cdot 2x = x^2 + 1 + 2x(x+2)$$.

Apply the Chain Rule

Now we will combine our results from Step 2 and Step 3 using the Chain Rule. The Chain Rule states that if \(y\) is a function of \(u\) and \(u\) is a function of \(x\), then the derivative of \(y\) with respect to \(x\) is: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$. Substituting the results from Steps 2 and 3, we get: $$\frac{dy}{dx} = 4u^{3} \cdot (x^2 + 1 + 2x(x+2))$$. Recall that \(u=(x+2)(x^2+1)\), so we will substitute this expression for \(u\): $$\frac{dy}{dx} = 4((x+2)(x^{2}+1))^{3}\cdot (x^2 + 1 + 2x(x+2))$$.

Simplify the Expression

Finally, we will simplify the expression: $$\frac{dy}{dx} = 4((x+2)(x^{2}+1))^{3}\cdot (x^2 + 3x^2 + 1 + 4x)$$. $$\frac{dy}{dx} = 4((x+2)(x^{2}+1))^{3}\cdot (4x^2 + 4x + 2)$$. So, the derivative of the given function is: $$\frac{dy}{dx} = 4((x+2)(x^{2}+1))^{3}(4x^2 + 4x + 2)$$.

Key concepts

Product Rule

Understanding the Product Rule is crucial for differentiating functions that are products of two or more functions. It states that the derivative of a product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function. Mathematically, if we have two functions, u(x) and v(x), their derivative can be represented as:
\[\begin{equation}\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\end{equation}\]
This technique was used in differentiating the inner function in our exercise, where we applied the Product Rule to \(u=(x+2)(x^2+1)\), treating \(x+2\) and \(x^{2}+1\) as separate functions.

Derivative of Polynomials

Polynomials are expressions made up of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The derivative of a polynomial function is relatively straightforward, and it's based on the power rule. For a term \(ax^n\), the derivative is found by multiplying the exponent by the coefficient and then subtracting one from the exponent:
\[\begin{equation}\frac{d}{dx}[ax^n] = nax^{n-1}\end{equation}\]
In our example, this rule helped us find the derivatives of \(\frac{d}{dx}(x+2)\) and \(\frac{d}{dx}(x^{2}+1)\) which are 1 and \(2x\), respectively.

Differentiation Techniques

There are several differentiation techniques each suited for different types of functions. These include the Power Rule, Product Rule, Quotient Rule, Chain Rule, and others. Each technique follows specific rules that fit the function it's being applied to. For instance, the Product Rule, as previously discussed, is used when dealing with the multiplication of functions. On the other hand, if the function was in the form of a fraction, we would use the Quotient Rule. It's important to choose the correct differentiation technique to effectively simplify the process and to obtain the correct derivative. The chosen technique should align with the nature of the function for optimal results.

Applying the Chain Rule

The Chain Rule is a powerful differentiation technique used for functions composed of other functions, known as composite functions. It allows us to differentiate the composite function by taking the derivative of the outer function and multiplying it by the derivative of the inner function. Symbolically, if \(y = f(g(x))\), then the Chain Rule tells us that:
\[\begin{equation}\frac{dy}{dx} = f'(g(x))g'(x)\end{equation}\]
In the given exercise, we used the Chain Rule to differentiate \(y = ((x+2)(x^2+1))^4\) by identifying \(u = (x+2)(x^2+1)\) as the inner function and then applying the Chain Rule to find \(\frac{dy}{dx}\). This was accomplished by differentiating \(u^4\) with respect to \(\frac{dy}{du}\) and then multiplying by the derivative of \(\frac{du}{dx}\), the latter of which was computed using the Product Rule, incorporating the derivatives of the polynomials within \(u\).