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Chapter 2: Problem 16

Use the Generalized Power Rule to find the derivative of each function. $$h(z)=\left(5 z^{2}+3 z-1\right)^{3}$$

Short Answer

Expert verified
The derivative of \( h(z) \) is \( h'(z) = 3(5z^2 + 3z - 1)^2(10z + 3) \).

Step by step solution

01

Identify the Functions Involved

Notice that the function given is a composition, specifically a power function applied to a polynomial function. We have an outer function \( g(u) = u^3 \) and an inner function \( u(z) = 5z^2 + 3z - 1 \). The goal is to differentiate \( h(z) = g(u(z)) \).
02

Apply the Generalized Power Rule

The Generalized Power Rule states that for a function \( (u(z))^n \), the derivative is \( n \cdot (u(z))^{n-1} \cdot u'(z) \). Applying this rule to our function \( h(z) = (5z^2 + 3z - 1)^3 \), we identify \( n = 3 \), which gives us the derivative \( 3 \cdot (5z^2 + 3z - 1)^2 \cdot u'(z) \).
03

Calculate the Derivative of the Inner Function

Now, find the derivative of the inner function \( u(z) = 5z^2 + 3z - 1 \). The derivative \( u'(z) \) is obtained by differentiating each term: \( \frac{d}{dz}[5z^2] = 10z \), \( \frac{d}{dz}[3z] = 3 \), and \( \frac{d}{dz}[-1] = 0 \). Thus, \( u'(z) = 10z + 3 \).
04

Substitute and Simplify

Substitute \( u'(z) \) back into the expression derived from Step 2: \( 3 \cdot (5z^2 + 3z - 1)^2 \cdot (10z + 3) \). Thus, the derivative of \( h(z) \) is \( h'(z) = 3(5z^2 + 3z - 1)^2(10z + 3) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative of Polynomial
Understanding how to find the derivative of a polynomial is fundamental in calculus. A polynomial is a mathematical expression made up of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. To differentiate a polynomial:
  • Apply the power rule: For any term with the form \( ax^n \), the derivative is \( nax^{n-1} \).
  • For constants, the derivative is zero.
Let's look at an example. For the inner function \( u(z) = 5z^2 + 3z - 1 \):
  • The derivative of \( 5z^2 \) is \( 10z \) because \( 2 \times 5 = 10 \).
  • The derivative of \( 3z \) is \( 3 \) because \( 1 \times 3 = 3 \).
  • The derivative of \( -1 \) is \( 0 \).
Thus, the total derivative of the polynomial \( u(z) \) is \( 10z + 3 \). Using these rules makes differentiating polynomials straightforward and crucial when working with more complex functions.
Composition of Functions
The concept of composition of functions is key when dealing with functions that are nested within each other, often seen in real-world applications. The composition \( h(z) = g(u(z)) \) means that you apply one function to the results of another. The inner function here is \( u(z) = 5z^2 + 3z - 1 \), and the outer is \( g(u) = u^3 \).
To find the derivative of a composite function, utilize the chain rule, a fundamental calculus technique:
  • Identify inner and outer functions.
  • Diferentiate the outer function with respect to the inner one, treating the inner function as a single variable.
  • Multiply by the derivative of the inner function.
Here, the derivative \( h'(z) \) is obtained not merely by the outer function's derivative, but by multiplying it with the derivative of \( u(z) \). This understanding is vital when dealing with multi-layered calculations.
Calculus Step by Step
Approaching calculus problems systematically is important for understanding and accuracy. Here's a detailed guide to handling problems with composition and derivatives:
  • Start by identifying all functions involved, noting which are inner and which are outer.
  • Use known rules like the power and chain rules for derivatives.
  • Calculate the derivative of each component step by step. For example, find \( u'(z) \) first.
  • Substitute back into your formula, ensuring you consider each component carefully.
In the example, after identifying \( h(z) = (5z^2 + 3z - 1)^3 \), we determined the derivative step by step: calculate \( u'(z) = 10z + 3 \), then apply the generalized power rule, resulting in \( 3 \, (5z^2 + 3z - 1)^2 \, (10z + 3) \). Step by step solutions aid in breaking down complex tasks into manageable parts, ensuring concepts are grasped methodically and thoroughly.

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Most popular questions from this chapter

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$\sqrt[3]{x^{3}+8}-5$$

Using your own words, explain in terms of instantaneous rates of change why the derivative is undefined where a function has a discontinuity.

If a bullet from a 9 -millimeter pistol is fired straight up from th ground, its height \(t\) seconds after it is fired will \(s(t)=-16 t^{2}+1280 t\) feet (neglecting air resistance) for \(0 \leq t \leq 80\). a. Find the velocity function. b. Find the time \(t\) when the bullet will be at its maximum height. [Hint: At its maximum height the bullet is moving neither up nor down, and has velocity zero. Therefore, find the time when the velocity \(v(t)\) equals zero.] c. Find the maximum height the bullet will reach. [Hint: Use the time found in part (b) together with the height function \(s(t) .]\)

Find the second derivative of each function. $$\frac{3 x+1}{3 x-1}$$

Use the Generalized Power Rule to find the derivative of each function. $$g(z)=2 z\left(3 z^{2}-z+1\right)^{4}$$
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