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Magazine Chapter 3: Problem 6
Find the derivative of the given function. $$ H(z)=\left(z^{3}-3 z^{2}+1\right)^{-3} $$
Short Answer
Expert verified
\( \frac{dH}{dz} = -3 (3z^{2} - 6z) (z^{3} - 3z^{2} + 1)^{-4} \)
Step by step solution
01
Identify the function and its form
The given function is \( H(z)=\left(z^{3}-3 z^{2}+1\right)^{-3} \). Notice that it is in the form of a power of a function.
02
Apply the chain rule
To find the derivative of a composite function, use the chain rule. The outer function is \( u^{-3} \) where \( u = z^{3}-3z^{2}+1 \). The chain rule states: \( \frac{dH}{dz} = \frac{d}{du} [u^{-3}] * \frac{du}{dz} \).
03
Differentiate the outer function
Differentiate the outer function with respect to \( u \): \( \frac{d}{du} [u^{-3}] = -3u^{-4} \).
04
Differentiate the inner function
Differentiate the inner function \( u = z^{3} - 3z^{2} + 1 \): \( \frac{du}{dz} = 3z^{2} - 6z \).
05
Combine the results
Combine the derivatives using the chain rule: \( \frac{dH}{dz} = -3u^{-4} * (3z^{2} - 6z) \). Substitute \( u = z^{3} - 3z^{2} + 1 \). This results in: \( \frac{dH}{dz} = -3(z^{3} - 3z^{2} + 1)^{-4} * (3z^{2} - 6z) \).
06
Simplify the expression
Simplify the final answer: \( \frac{dH}{dz} = -3 (3z^{2} - 6z) (z^{3} - 3z^{2} + 1)^{-4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Chain Rule
The chain rule is a fundamental tool in calculus used to differentiate composite functions. A composite function is one where a function is nested inside another function, like in the given problem. For example, if you have a function nested inside another like \(H(z) = (z^3 - 3z^2 + 1)^{-3}\), you need the chain rule for differentiation.
To apply the chain rule, follow these steps:
\(\frac{dH}{dz} = \frac{d}{du} [u^{-3}] * \frac{du}{dz}\), where the inner \(u\) is \(z^3 - 3z^2 + 1\). The application of the chain rule helps handle complex differentiations efficiently.
To apply the chain rule, follow these steps:
- Identify the outer and inner functions. Here, the outer function is \(u^{-3}\) and the inner function is \(u = z^3 - 3z^2 + 1\).
- Differentiate the outer function with respect to the inner function.
- Differentiate the inner function concerning the main variable.
- Multiply these derivatives to get the final derivative using the chain rule.
\(\frac{dH}{dz} = \frac{d}{du} [u^{-3}] * \frac{du}{dz}\), where the inner \(u\) is \(z^3 - 3z^2 + 1\). The application of the chain rule helps handle complex differentiations efficiently.
Differentiation: Finding the Derivative
Differentiation is the process of calculating a derivative, which represents how a function changes as its input changes. The derivative gives us the rate of change or the slope of a function at any point. For the function \(H(z)\), to find \(\frac{dH}{dz}\), we use the chain rule.
Here's the step-by-step process:
Here's the step-by-step process:
- First, identify the inner and outer functions. For \(H(z) = (z^3 - 3z^2 + 1)^{-3}\), the outer function is \(u^{-3}\) and the inner function is \(u = z^3 - 3z^2 + 1\).
- Differentiate the outer function with respect to the inner function: \(\frac{d}{du} [u^{-3}] = -3u^{-4}\).
- Next, differentiate the inner function with respect to \(z\): \(\frac{du}{dz} = 3z^2 - 6z\).
- Now, multiply these derivatives: \(\frac{dH}{dz} = -3u^{-4} * (3z^2 - 6z)\).
- Replace \(u\) with \(z^3 - 3z^2 + 1\) to get the simplified derivative: \(\frac{dH}{dz} = -3 (3z^2 - 6z) (z^3 - 3z^2 + 1)^{-4}\).
Composite Functions: Functions Within Functions
Composite functions are functions composed of other functions. For example, if \(h(x) = f(g(x))\), then \(h(x)\) is a composite function. The given function \(H(z) = (z^3 - 3z^2 + 1)^{-3}\) is a composite function where \((z^3 - 3z^2 + 1)\) is nested inside the outer function \(u^{-3}\).
Understanding the parts of a composite function helps in applying the chain rule for differentiation.
Understanding the parts of a composite function helps in applying the chain rule for differentiation.
- Identify the inner function: \(u = z^3 - 3z^2 + 1\).
- Identify the outer function: \(u^{-3}\).
- Apply the chain rule to differentiate by treating the inner function as a single entity during the differentiation of the outer function.
