Problem 25
Question
Use the Chain Rule-Power Rule to differentiate the given expression with respect to \(x\). $$ 12 \sqrt{1+e^{x}} $$
Step-by-Step Solution
Verified Answer
The derivative is \( 6e^{x}(1 + e^{x})^{-1/2} \).
1Step 1: Identify Outer and Inner Functions
The given function is structured as \( 12 \sqrt{1 + e^{x}} \). We can express this as \( 12(1 + e^{x})^{1/2} \) to facilitate differentiation. The outer function is \( u^{1/2} \) where \( u = 1 + e^{x} \), and the inner function is \( 1 + e^{x} \).
2Step 2: Differentiate the Outer Function
Apply the power rule to differentiate the outer function \( u^{1/2} \). The derivative of \( u^{n} \) is \( n imes u^{n-1} \). Thus, the derivative of \( (1 + e^{x})^{1/2} \) with respect to \( u \) is \( \frac{1}{2}(1 + e^{x})^{-1/2} \).
3Step 3: Differentiate the Inner Function
Now, differentiate the inner function \( 1 + e^{x} \) with respect to \( x \). The derivative of a constant is 0, and the derivative of \( e^{x} \) is \( e^{x} \). Therefore, the derivative is \( e^{x} \).
4Step 4: Apply the Chain Rule
To apply the chain rule, multiply the derivative of the outer function by the derivative of the inner function. This results in: \( \frac{1}{2}(1 + e^{x})^{-1/2} \times e^{x} \).
5Step 5: Simplify the Expression
Finally, simplify the expression and multiply it by the constant factor. The derivative of the original function with respect to \( x \) is: \( 12 \times \frac{1}{2} \times (1 + e^{x})^{-1/2} \times e^{x} = 6 \times e^{x} \times (1 + e^{x})^{-1/2} \).
Key Concepts
Power RuleDifferentiationInner FunctionOuter Function
Power Rule
The Power Rule is a basic technique used in calculus to find the derivative of functions raised to a power. When you have an expression like \( u^n \), where \( n \) is a constant, the Power Rule helps you find its derivative efficiently.
The rule states that the derivative of \( u^n \) is \( nu^{n-1} \).
This rule allows us to easily handle powers without separately computing each element's derivative.
The rule states that the derivative of \( u^n \) is \( nu^{n-1} \).
This rule allows us to easily handle powers without separately computing each element's derivative.
- For example, if you have an expression \((1 + e^x)^{1/2}\), applying the Power Rule tells us the derivative with respect to \( u \) is \( \frac{1}{2}(1 + e^x)^{-1/2} \).
- This effectively reduces the exponent by one, and multiplies the whole by the existing exponent.
Differentiation
Differentiation is a fundamental concept in calculus, which involves finding the rate at which a function changes at any given point. This rate of change is called the derivative.
The process handles diverse types of functions, but fundamentally boils down to determining slopes of curves.
The process handles diverse types of functions, but fundamentally boils down to determining slopes of curves.
- In the context of our exercise, differentiation allows us to find out how the function \( 12 \sqrt{1+e^{x}} \) behaves as \( x \) varies.
- By using rules like the Power Rule and Chain Rule, we systematically dissect the function into parts that can be easily differentiated.
Inner Function
The concept of an Inner Function is crucial when dealing with composite functions, which are functions encapsulated within others.
In our exercise, the inner function is \( 1 + e^x \). It is the core expression we differentiate directly.
In our exercise, the inner function is \( 1 + e^x \). It is the core expression we differentiate directly.
- Identifying the inner function assists in structuring our differentiation strategy, especially when applying the Chain Rule.
- The inner function often sets the stage for uncovering how changes in \( x \) affect the more complex structure as a whole.
- Through differentiation, the inner function's role clarifies, as its derivative, \( e^x \), becomes a crucial factor in the final derivative.
Outer Function
The Outer Function represents the broader structure containing the inner core of a composite function.
In the given example, the outer function is \( u^{1/2} \) where \( u = 1 + e^x \).
It permits focusing on one layer at a time without losing sight of the entire function's behavior.
In the given example, the outer function is \( u^{1/2} \) where \( u = 1 + e^x \).
- Once you have identified the inner function, the outer function takes center stage to accommodate the overall architecture of the function.
- The outer structure informs how the final expression should be differentiated in stages, with derivatives stacking up via mechanisms like Power Rule.
It permits focusing on one layer at a time without losing sight of the entire function's behavior.
Other exercises in this chapter
Problem 25
Use implicit differentiation to find the normal line to the given curve at the given point \(P_{0}\).\(x y^{4}-x^{3} y=16 \quad P_{0}=(2,2)\)
View solution Problem 25
An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(\log _{3}(x)\)
View solution Problem 25
Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(-\infty, \infty) \rightarrow(0, \infty), f(s)=2^{s} $$
View solution Problem 25
Use the Reciprocal Rule to compute the derivative of the given expression with respect to \(x\) $$ 9 /\left(3 \cos (x)+x^{3}\right) $$
View solution