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Chapter 3: Problem 71

Find the first and second derivatives. $$ k(r)=(4 r+7)^{5} $$

Short Answer

Expert verified
The first derivative is \(20(4r+7)^4\) and the second derivative is \(320(4r+7)^3\).

Step by step solution

01

Identify the Function Form

The given function is a composite function in the form of \(k(r) = (u(r))^5\), where \(u(r) = 4r + 7\). This indicates that we'll need to use the chain rule for differentiation.
02

Differentiate Using the Chain Rule

To find the first derivative \(k'(r)\), use the chain rule: \(k'(r) =\frac{d}{dr}[(u(r))^5] = 5(u(r))^4 \cdot u'(r)\). First, find \(u'(r)\).
03

Differentiate the Inner Function

Find the derivative of \(u(r) = 4r + 7\), which is \(u'(r) = 4\).
04

Apply the Chain Rule

Insert \(u'(r)\) into the chain rule: \(k'(r) = 5 (4r + 7)^4 \cdot 4 = 20(4r + 7)^4\). This is the first derivative of the function.
05

Differentiate the First Derivative

To find the second derivative, differentiate the first derivative \(k'(r) = 20(4r+7)^4\) using the chain rule again.
06

Apply the Chain Rule to the First Derivative

The function inside the first derivative is \(v(r) = (4r+7)^4\). Differentiate: \(v'(r) = 4(4r+7)^3 \cdot 4 = 16(4r+7)^3\).
07

Find the Second Derivative

The second derivative \(k''(r)\) is given by differentiating \(k'(r)\): \(k''(r) = 20 \cdot 16 (4r+7)^3 \cdot 4 = 320(4r+7)^3\).

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Key Concepts

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

Chain Rule in Calculus
The chain rule is a fundamental technique in calculus used for finding the derivative of a composite function. A composite function is essentially a function within another function, such as \((4r + 7)^5\). Here, the expression \(4r + 7\) is inside the function being raised to the power of 5. This is why we use the chain rule.The chain rule states that if you have a function \(h(x) = f(g(x))\), the derivative \(h'(x)\) is given by \(f'(g(x)) \cdot g'(x)\).
  • Began by differentiating the outer function, keeping the inner function the same.
  • Then, multiply by the derivative of the inner function.
For \((4r + 7)^5\), we first differentiate the outer power function, resulting in \(5(4r + 7)^4\). Then, multiply by \(4\), the derivative of the inner function \(4r + 7\). This results in the first derivative \(20(4r + 7)^4\).
First Derivative
The first derivative of a function provides information about the rate of change of the function at any point. When we differentiate a function, we are effectively finding the slope of the tangent line at any given point of the function.In our exercise with \(k(r) = (4r + 7)^5\), the first step is to use the chain rule. We begin by differentiating the entire expression as if \((4r + 7)\) is a single entity:
  • The derivative of \((u(r))^5\) with respect to \(u(r)\) is \(5(u(r))^4\).
  • Then, differentiate \(u(r) = 4r + 7\), giving us \(4\).
  • Thus, the first derivative \(k'(r)\) becomes \(20(4r + 7)^4\).
This derivative tells us how \(k(r)\) changes with a small increase or decrease in \(r\).
Second Derivative
The second derivative of a function gives us insights into the concavity of the graph and whether it is curving upwards or downwards. It also plays a role in identifying inflection points, where the curve changes its concavity.To find the second derivative from our first derivative \(k'(r) = 20(4r + 7)^4\), we'll once again apply the chain rule. Here鈥檚 the process:
  • Identify the new inner function \(v(r) = (4r + 7)^4\).
  • Differentiating \(v(r)\) gives \(16(4r + 7)^3\).
  • Multiply this result by the constant from the first derivative, giving us \(320(4r + 7)^3\).
This tells us how the rate of change itself is changing. A positive second derivative indicates a concave up curve, while a negative one indicates a concave down curve.
Composite Function
A composite function occurs when one function is nested inside another function, like our example \(k(r) = (4r + 7)^5\). Understanding the nature of composite functions is essential when determining derivatives, as you often need to use techniques like the chain rule.In the given function, \(4r + 7\) is the inner function, and raising this expression to the power of 5 is the outer function. We handle these separately when differentiating:
  • View the entire inside expression as a single variable when differentiating the outer function.
  • Then, perform the derivative of the inner part.
By decomposing the function into its components, you can systematically apply the chain rule and other derivative techniques. Recognizing composite functions is crucial for tackling complex calculus problems effectively.

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Most popular questions from this chapter

Exer. \(43-44\) : Given the position function \(s\) of a point \(P\) moving on a coordinate line \(l\), find the times at which the velocity is the given value \(k\). $$ s(t)=3 t^{2 / 3} ; \quad k=4 $$

If \(y=3+2 \sin x,\) find (a) the \(x\) -coordinates of all points on the graph at which the tangent line is parallel to the line \(y=\sqrt{2} x-5\) (b) an equation of the tangent line to the graph at the point on the graph with \(x\) -coordinate \(\pi / 6\)

Find \(y^{\prime}, y^{\prime \prime},\) and \(y^{\prime \prime \prime}\). $$ y=5 x^{3}+4 \sqrt{x} $$

Assuming that the equation determines a differentiable function \(f\) such that \(y=f(x),\) find \(y^{\prime}\). $$ \frac{\sqrt{x}+1}{\sqrt{y}+1}=y $$

If \(k(x)=f(g(x))\) and if \(f(2)=-4, g(2)=2, f^{\prime}(2)=3\), and \(g^{\prime}(2)=5\), find \(k(2)\) and \(k^{\prime}(2)\).
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