Question

Derivatives Find and simplify the derivative of the following functions. \(f(x)=\sqrt{e^{2 x}+8 x^{2} e^{x}+16 x^{4}}\) (Hint: Factor the function under the square root first.)

Short answer

Answer: The derivative of the given function is \(f'(x) = 2(e^{x} + 4x^{2})(e^{x} + 8x)\).

Step-by-step solution

Factor the function under the square root

First, we will factor the function under the square root: \(e^{2 x}+8 x^{2} e^{x}+16 x^{4} = e^{x}(e^{x} + 8 x^2) + 16x^4\) Notice that \(e^{x}\) is common in first two terms and can be factored out further: \(e^{2 x}+8 x^{2} e^{x}+16 x^{4} = (e^{x} + 4x^2)(e^{x} + 4x^2)\)

Rewrite the function with factored form

Now we rewrite the function \(f(x)\) using the factored form under the square root: \(f(x) = \sqrt{(e^{x} + 4x^{2})(e^{x} + 4x^{2})} = \sqrt{(e^{x} + 4x^{2})^2}\)

Apply the chain rule

Now we apply the chain rule to find the derivative of \(f(x)\): \(f'(x) = \frac{1}{2} [(e^{x} + 4x^{2})^2]^{-\frac{1}{2}} \cdot (2(e^{x} + 4x^{2})(e^{x} + 8x))\)

Simplify the derivative

Simplify the expression for the derivative of \(f(x)\): \(f'(x) = \frac{1}{\sqrt{(e^{x} + 4x^{2})^2}} \cdot (2(e^{x} + 4x^{2})(e^{x} + 8x))\) Since we have \({(e^{x} + 4x^{2})^2}\) in the denominator and square root of it, we can simplify the expression further: \(f'(x) = (2(e^{x} + 4x^{2})(e^{x} + 8x))\) Therefore, the derivative of the given function is \(f'(x) = 2(e^{x} + 4x^{2})(e^{x} + 8x)\).

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus for taking the derivative of composite functions. If you have a function that can be written as a composition of two or more functions, the chain rule helps you find its derivative. In simple terms, it states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This powerful rule is often summarized by the formula: \[(f(g(x)))' = f'(g(x)) \cdot g'(x)\] In the given exercise, the function \(f(x) = \sqrt{(e^{x} + 4x^{2})^{2}}\) falls neatly into the chain rule setup. The outer function is the square root \((x^{2})^{1/2}\) and the inner function is \((e^{x} + 4x^2)^{2}\). By the chain rule, the derivative is found by:

• First taking the derivative of the outer function, appending the expression inside it.
• Then, multiplying it by the derivative of the inner function.

This step of using the chain rule allows us to break down complex expressions into manageable derivatives.

Factoring Algebraic Expressions

Factoring is a critical skill in simplifying expressions and solving equations both in algebra and calculus. The process involves rewriting a mathematical expression as a product of its simpler parts, or 'factors'. Recognizing common factors allows us to rewrite the expressions in a form that is often easier to manipulate. For the given problem, the aim was to factor the expression inside the square root: \[e^{2 x}+8 x^{2} e^{x}+16 x^{4}\] Upon inspection, the terms \(e^{2x}\) and \(8x^{2}e^{x}\) share a common factor of \(e^{x}\), allowing us to express the original expression as: \[e^{x}(e^{x} + 8x^{2}) + 16x^{4}\]Recognizing this shared factor helps to simplify to the form \((e^{x} + 4x^{2})^{2}\). This form is particularly useful as it appears squared under the square root, effectively tipping us off to further simplification.

Simplifying Expressions

Simplifying expressions is the act of transforming a complex expression into a more manageable form. This can often involve combining like terms, reducing coefficients, and canceling terms when applicable. Simplification is particularly useful when solving differential equations or integrating functions.In this exercise, after applying the chain rule, it’s necessary to simplify the derivative expression to reach the final answer. The original form of the derivative was: \[\frac{1}{\sqrt{(e^{x} + 4x^{2})^2}} \cdot (2(e^{x} + 4x^{2})(e^{x} + 8x))\] Notice that the denominator \(\sqrt{(e^{x} + 4x^{2})^2}\) can be reduced since the square root cancels out the square, simplifying it to just \((e^{x} + 4x^{2})\). By performing this simplification, the expression can be broken down further:

• Cancel the term \((e^{x} + 4x^{2})\) in the numerator and the denominator.
• Leaving just the simplified form \(2(e^{x} + 8x)\).

This streamlined expression is much easier to work with for any subsequent calculations or evaluations.