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Chapter 3: Problem 6

Find the derivatives of the functions. \(\sqrt{625-x^{2}}+3 x^{3}+12\)

Short Answer

Expert verified
The derivative of the function is \( f'(x) = -\frac{x}{\sqrt{625-x^{2}}} + 9x^{2} \).

Step by step solution

01

Identify the Function Components

The function given is \( f(x) = \sqrt{625-x^{2}} + 3x^{3} + 12 \). This function contains three parts: \( \sqrt{625-x^{2}} \), \( 3x^{3} \), and the constant \( 12 \).
02

Differentiate the First Component

To differentiate \( \sqrt{625-x^{2}} \), rewrite it as \( (625-x^{2})^{1/2} \). Use the chain rule: The derivative of \( u^{n} \) is \( nu^{n-1} \cdot u' \). Here, \( u = 625-x^{2} \) and \( n = 1/2 \). So, \( u' = -2x \). The derivative is \(-x(625-x^{2})^{-1/2} \).
03

Differentiate the Second Component

The second component is \( 3x^{3} \). Use the power rule for derivatives: The derivative of \( cx^{n} \) is \( cnx^{n-1} \). Therefore, the derivative of \( 3x^{3} \) is \( 9x^{2} \).
04

Differentiate the Third Component

The third component is a constant, \( 12 \). The derivative of any constant is \( 0 \).
05

Combine the Derivatives

Combine the derivatives from the previous steps to find the derivative of the entire function: \( f'(x) = -x(625-x^{2})^{-1/2} + 9x^{2} + 0 \). Thus, the derivative simplifies to \( f'(x) = -\frac{x}{\sqrt{625-x^{2}}} + 9x^{2} \).

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Key Concepts

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

Chain Rule
The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. A composite function is essentially a function within another function, and the chain rule helps break down these layers to find the derivative.
  • To use the chain rule, identify the outer and inner functions. In our exercise, \( \sqrt{625-x^{2}} \) can be rewritten as \( (625-x^{2})^{1/2} \), where \( u = 625-x^{2} \) is the inner function.
  • The outer function in this setup is \( u^{1/2} \). The chain rule formula is \( \frac{d}{dx}[u^{n}] = nu^{n-1} \cdot u' \).
  • First, differentiate the outer function: \( \frac{d}{dx}[u^{1/2}] = \frac{1}{2}u^{-1/2} \).
  • Then, differentiate the inner function (\( u = 625-x^{2} \)): \( u' = -2x \).
  • Combine these results: The derivative of the entire composite function is \( -x(625-x^{2})^{-1/2} \).
Understanding and applying the chain rule involves recognizing these components and systematically practicing to gain proficiency.
Power Rule
The power rule is one of the easiest and most frequently used rules for finding derivatives. It simplifies the process by providing a straightforward method for differentiating functions of the form \( cx^{n} \).
  • If you have a function \( f(x) = cx^{n} \), the power rule states that its derivative is \( f'(x) = cnx^{n-1} \).
  • For the exercise at hand, consider the component \( 3x^{3} \). According to the power rule, differentiate it as follows: \( 3 \cdot 3x^{3-1} \).
  • This gives us the derivative \( 9x^{2} \).
  • Notice how the power of \( x \) is reduced by one while the coefficient is multiplied by the original exponent.
The power rule is thus a quick and efficient way to tackle polynomial expressions when taking derivatives.
Derivative of a Constant
When it comes to derivatives, constants have a special rule of their own: they simply vanish. This is because the derivative measures the rate of change, and a constant, by its nature, doesn't change.
  • If you have a constant value like \( c \), its derivative is always \( 0 \).
  • For example, in our function, we have the constant term \( 12 \). Its derivative is \( 0 \) because it doesn't change as \( x \) changes.
  • This simplification often helps to streamline the differentiation process by eliminating any constant terms immediately.
By understanding this fundamental concept, we can quickly handle constants, allowing us to focus on the variable parts of the function.

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Most popular questions from this chapter

Find the derivatives of the functions using the quotient rule. \(\frac{x^{2}+5 x-3}{x^{5}-6 x^{3}+3 x^{2}-7 x+1}\)

If \(f^{\prime}(4)=5, g^{\prime}(4)=12,(f g)(4)=f(4) g(4)=2,\) and \(g(4)=6,\) compute \(f(4)\) and \(\frac{d}{d x} \frac{f}{g}\) at \(4 .\)

The general polynomial \(P\) of degree \(n\) in the variable \(x\) has the form \(P(x)=\sum_{k=0}^{n} a_{k} x^{k}=\) \(a_{0}+a_{1} x+\ldots+a_{n} x^{n}\). What is the derivative (with respect to \(x\) ) of $P ?

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