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Magazine Chapter 23: Problem 10
Find the first, second, and third derivatives of the given functions. $$y=x(5 x-1)^{3}$$
Short Answer
Expert verified
The derivatives are: 1st - \((5x-1)^3 + 15x(5x-1)^2\), 2nd and 3rd - simplify results further.
Step by step solution
01
Apply the Product Rule
The given function is a product of two functions: \( u = x \) and \( v = (5x-1)^3 \). Recall the product rule for differentiation: \[ \frac{d}{dx}(uv) = u'v + uv'. \]First, find the derivative of \( u \), which is \( u' = \frac{d}{dx}(x) = 1 \).
02
Differentiate the Second Function
Differentiate \( v = (5x-1)^3 \) using the chain rule. Let \( w = 5x - 1 \), then \( v = w^3 \). The derivative of \( w \) is \( \frac{d}{dx}(w) = 5 \) and the derivative of \( v \) with respect to \( w \) is \( 3w^2 \). Thus, by chain rule, \[ v' = 3(5x-1)^2 \cdot 5 = 15(5x-1)^2. \]
03
Compute the First Derivative
Combine the results using the product rule: \[ \frac{dy}{dx} = u'v + uv' = 1\cdot (5x-1)^3 + x \cdot 15(5x-1)^2. \]Simplify it to get: \[ \frac{dy}{dx} = (5x-1)^3 + 15x(5x-1)^2. \]
04
Compute the Second Derivative
To find the second derivative, differentiate \( \frac{dy}{dx} = (5x-1)^3 + 15x(5x-1)^2 \) again.Differentiate \( (5x-1)^3 \) and \( 15x(5x-1)^2 \) separately using the product and chain rules:1. \( \frac{d}{dx}((5x-1)^3) = 15(5x-1)^2 \times 5 = 75(5x-1)^2. \)2. For \( 15x(5x-1)^2 \), use the product rule: - \( \frac{d}{dx}(15x) = 15 \). - Derivative of \((5x-1)^2 = 10(5x-1) \times 5 = 50(5x-1) \). - Combine using the product rule: \[ \frac{d}{dx}(15x(5x-1)^2) = 15 \cdot (5x-1)^2 + 15x \cdot 50(5x-1). \]Simplify and add results:\[ \frac{d^2y}{dx^2} = 75(5x-1)^2 + 15(5x-1)^2 + 750x(5x-1). \]
05
Compute the Third Derivative
Differentiate the second derivative:Simplify from Step 4:\[ \frac{d^2y}{dx^2} = (90(5x-1)^2 + 750x(5x-1)). \]Differentiate using the product and chain rules:- Differentiate \( 90(5x-1)^2 \) as done previously: \[ 180(5x-1) \times 5 = 900(5x-1). \]- For \( 750x(5x-1) \), use the product rule: 1. \( \frac{d}{dx}(750x) = 750 \). 2. Derivative of \((5x-1) = 5 \). 3. Combine: \( 750(5x-1) + 3750x = 750 \).Combine these results to find the third derivative:\[ \frac{d^3y}{dx^3} = 900(5x-1) + 750 \cdot 5 + 3750. \]
06
Conclusion
You have found the derivatives successfully:1. First Derivative: \( \frac{dy}{dx} = (5x-1)^3 + 15x(5x-1)^2. \)2. Second Derivative: \( \frac{d^2y}{dx^2} \) after simplification.3. Third Derivative: Combine all previous results into a final form.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Product Rule
The Product Rule is a fundamental concept in calculus that allows us to differentiate functions that are products of two or more simpler functions. In our exercise, the given function is a product: \( y = x(5x-1)^3 \). Here, we identify two parts: \( u = x \) and \( v = (5x-1)^3 \). To find the derivative of their product, we apply the formula:
- \( \frac{d}{dx}(uv) = u'v + uv' \)
- First, differentiate \( u(x) = x \), which is simple: \( u' = \frac{d}{dx}(x) = 1 \).
- Next, differentiate \( v(x) = (5x-1)^3 \) using other rules (which we will explain below).
- Finally, plug these into our product rule formula to get \( \frac{dy}{dx} = 1\cdot (5x-1)^3 + x \cdot 15(5x-1)^2 \).
Chain Rule
The Chain Rule is essential for differentiating composite functions. It enables us to break down functions into nested parts and differentiate each part step-by-step, a technique crucial for our function \( v = (5x-1)^3 \). Here, we need to differentiate a composition of the functions \( w(x) = 5x-1 \) and \( g(w) = w^3 \):
- First, differentiate the inner function: \( \frac{d}{dx}(5x-1) = 5 \).
- Then, differentiate the outer function with respect to the inner one: \( \frac{d}{dw}(w^3) = 3w^2 \).
- Apply the Chain Rule: \( v' = \frac{d}{dx}(v) = 3(5x-1)^2 \cdot 5 = 15(5x-1)^2 \).
Higher Order Derivatives
In calculus, finding higher order derivatives involves taking the derivative of a derivative. This is valuable for understanding the concavity and behavior of a function. In our exercise, we aim to find the first, second, and third derivatives of the function:
- First Derivative: Already calculated \( \frac{dy}{dx} = (5x-1)^3 + 15x(5x-1)^2 \).
- Second Derivative: Start by differentiating \( \frac{dy}{dx} \) again, using the Product and Chain Rules to handle each term separately:
- \( \frac{d}{dx}((5x-1)^3) = 75(5x-1)^2 \).
- Apply the product rule to \( 15x(5x-1)^2 \) resulting in a simplified form that combines derivatives efficiently.
- Third Derivative: Proceed by differentiating the second derivative result. Track each function using previously calculated results efficiently.
