Problem 51
Question
In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ \frac{d}{d z}(1+(F(2 z)))^{2} $$
Step-by-Step Solution
Verified Answer
The derivative is \(4(1 + F(2z))F'(2z)\).
1Step 1: Identify the Composite Function
Here we have a composite function, which is of the form \((u(z))^2\) where \(u(z) = 1 + F(2z)\). We need to differentiate it with respect to \(z\).
2Step 2: Apply the Chain Rule
According to the chain rule, if you have a function \((u(z))^n\), its derivative with respect to \(z\) is \(n(u(z))^{n-1} \cdot u'(z)\). For our problem, \(n = 2\) and \(u(z) = 1 + F(2z)\).
3Step 3: Differentiate the Outer Function
Apply the power rule to differentiate \((u(z))^2\). The derivative is \(2(u(z))(u'(z))\). So, we have:\[2(1 + F(2z))u'(z)\].
4Step 4: Differentiate the Inner Function
Differentiate \(u(z) = 1 + F(2z)\) with respect to \(z\). The derivative \(u'(z)\) is \(F'(2z) \cdot 2\) because applying the chain rule on \(F(2z)\) gives \(F'(2z)\times \text{derivative of }(2z)\), which is \(2\).
5Step 5: Assemble the Derivative Expression
Substitute \(u'(z)\) back into the expression from Step 3, giving:\[2(1 + F(2z))(F'(2z) \cdot 2)\].Simplify it to:\[4(1 + F(2z))F'(2z)\].
Key Concepts
Composite Function DifferentiationPower RuleDifferentiating Composite FunctionsCalculus Derivatives
Composite Function Differentiation
When you're dealing with a composite function, you're working with a function inside another function. In this exercise, the expression \((1 + F(2z))^2\) is our composite function. It involves two layers: one, the inner function \(F(2z)\), and two, the outer function raised to the power of 2. Differentiating composite functions requires understanding how these layers interact.
When differentiating such functions, the chain rule becomes your best friend. This rule helps by allowing you to take the derivative of the outer layer and multiply it by the derivative of the inner layer. Recognizing that you have a composite function is the first vital step in applying the chain rule effectively.
When differentiating such functions, the chain rule becomes your best friend. This rule helps by allowing you to take the derivative of the outer layer and multiply it by the derivative of the inner layer. Recognizing that you have a composite function is the first vital step in applying the chain rule effectively.
Power Rule
The power rule is a fundamental principle in calculus for differentiating expressions of the form \(x^n\), where \(n\) is a constant. In our exercise, the expression \((1 + F(2z))^2\) uses the power rule to differentiate the outer function.
According to the power rule, if you have \((u(z))^n\), its derivative is \(n\cdot (u(z))^{n-1}\). Here, \(n = 2\), and \(u(z)\) represents \(1 + F(2z)\). So first, you differentiate \((1 + F(2z))^2\) to get \(2(1 + F(2z))\). Keep in mind, though, this isn't the final derivative but just a part of the process, as you need to apply the chain rule next to consider the inner function.
According to the power rule, if you have \((u(z))^n\), its derivative is \(n\cdot (u(z))^{n-1}\). Here, \(n = 2\), and \(u(z)\) represents \(1 + F(2z)\). So first, you differentiate \((1 + F(2z))^2\) to get \(2(1 + F(2z))\). Keep in mind, though, this isn't the final derivative but just a part of the process, as you need to apply the chain rule next to consider the inner function.
Differentiating Composite Functions
After using the power rule to tackle the outer function, you're ready to differentiate the inner function with the chain rule.
In our example, \(u(z) = 1 + F(2z)\). Differentiating \(u(z)\) means finding \(u'(z)\). Since \(1\) is constant, its derivative is zero, leaving only \(F(2z)\) to differentiate. Applying the chain rule, differentiate \(F(2z)\) as \(F'(2z)\) and multiply by the derivative of \(2z\), which is \(2\). This gives you \(u'(z) = 2F'(2z)\).
This result, \(2F'(2z)\), is the inner function's contribution to the whole derivative.
In our example, \(u(z) = 1 + F(2z)\). Differentiating \(u(z)\) means finding \(u'(z)\). Since \(1\) is constant, its derivative is zero, leaving only \(F(2z)\) to differentiate. Applying the chain rule, differentiate \(F(2z)\) as \(F'(2z)\) and multiply by the derivative of \(2z\), which is \(2\). This gives you \(u'(z) = 2F'(2z)\).
This result, \(2F'(2z)\), is the inner function's contribution to the whole derivative.
Calculus Derivatives
Understanding derivatives is at the heart of calculus. A derivative measures how a function's output changes as its input changes. The process involves several rules, like the power rule and the chain rule, allowing us to tackle different expressions.
For the composite function \((1 + F(2z))^2\), we combined these rules to find the derivative. First, we differentiated the outer layer, applying the power rule, then we focused on the inner layer using the chain rule. The culmination of these steps yields the final derivative:
For the composite function \((1 + F(2z))^2\), we combined these rules to find the derivative. First, we differentiated the outer layer, applying the power rule, then we focused on the inner layer using the chain rule. The culmination of these steps yields the final derivative:
- Start with the power rule: \(2(1 + F(2z))\)
- Incorporate the derivative from the chain rule: \(2F'(2z)\)
Other exercises in this chapter
Problem 51
In Problems 51-56, first find and simplify $$ \frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ Then find \(d y / d x\) by taking the limit of yo
View solution Problem 51
Find all points on the graph of \(y=x^{3}-x^{2}\) where the tangent line is horizontal.
View solution Problem 52
Find the indicated derivative. \(D_{x} x^{\left(2^{x}\right)}\)
View solution Problem 52
\(y=x^{3}-3 x^{2}\)
View solution