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Chapter 27: Problem 9

Find the derivatives of the given functions. $$y=4 x^{3}-3 \csc \sqrt{2 x+3}$$

Short Answer

Expert verified
The derivative is \( y' = 12x^{2} + 3 \cdot \csc(\sqrt{2x+3}) \cdot \cot(\sqrt{2x+3}) \cdot (2x+3)^{-1/2} \).

Step by step solution

01

Analyze the Function

The function given is a combination of a polynomial and trigonometric expression: \( y = 4x^{3} - 3 \csc(\sqrt{2x+3}) \). To find the derivative, we need to apply basic differentiation rules, including the power rule and chain rule.
02

Apply the Power Rule

For the term \( 4x^{3} \), use the power rule: if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \). Therefore, the derivative of \( 4x^{3} \) is \( 12x^{2} \).
03

Differentiate the Cosecant Term Using Chain Rule

For the term \(-3 \csc(\sqrt{2x+3})\), differentiate \(-\csc(u)\) using the chain rule. First, remember that the derivative of \( \csc(u) \) is \(-\csc(u)\cot(u)\). Let \( u = \sqrt{2x+3} \).
04

Find the Derivative of Inner Function

Find the derivative of \( u = \sqrt{2x+3} \). Rewriting \( u \) as \( (2x+3)^{1/2} \), its derivative is \( u' = \frac{1}{2} (2x+3)^{-1/2} \cdot 2 = (2x+3)^{-1/2} \).
05

Combine Derivatives Using Product Rule and Chain Rule

The derivative of \(-3\csc(\sqrt{2x+3})\) is \( 3 \cdot \csc(\sqrt{2x+3})\cdot \cot(\sqrt{2x+3})\cdot (2x+3)^{-1/2} \). This result is derived by applying the chain rule to the outside function \( -\csc \) and the inside function \( \sqrt{2x+3} \).
06

Final Combination of Derivatives

Combine the derivatives from steps 2 and 5 to get the final derivative: \[ y' = 12x^{2} + 3 \cdot \csc(\sqrt{2x+3}) \cdot \cot(\sqrt{2x+3}) \cdot (2x+3)^{-1/2} \].

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Key Concepts

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

Power Rule
The power rule is a cornerstone concept in calculus, especially when dealing with polynomial functions. It is a straightforward differentiation technique that simplifies the process of finding derivatives of power functions. The power rule states:
  • If you have a function in the form of \( f(x) = ax^n \), where \( a \) is a constant, and \( n \) is a real number, the derivative \( f'(x) \) is obtained by multiplying the exponent by the coefficient and then subtracting one from the exponent. So the rule is \( f'(x) = anx^{n-1} \).
Applying the power rule to the term \( 4x^3 \) from our example function, we differentiate it to find its derivative:
  • Multiply the coefficient (4) by the exponent (3): \( 4 \times 3 = 12 \)
  • Subtract 1 from the exponent (3): \( 3 - 1 = 2 \)
  • The derivative is \( 12x^{2} \).
This process is quick and efficient, making it a valuable tool for students when working with polynomial expressions.
Chain Rule
The chain rule is an essential technique for finding derivatives of composite functions. A composite function is a function within another function, like \( g(f(x)) \). The chain rule helps us differentiate such functions by focusing on the outer and the inner functions separately.
  • First, differentiate the outer function, keeping the inner function unchanged.
  • Next, multiply the result by the derivative of the inner function.
Consider the function \(-3 \csc(\sqrt{2x+3}) \). Here, the outer function is \(- \csc(u)\), where \( u = \sqrt{2x+3} \). The derivative of \(- \csc(u)\) is \(- \csc(u) \cot(u)\). Apply the chain rule by then multiplying by the derivative of the inner function \( u \):
  • The derivative of \( u = \sqrt{2x+3} \) can be found by rewriting it as \( (2x+3)^{1/2} \).
  • Differentiating \( (2x+3)^{1/2} \), we get \((2x+3)^{-1/2} \cdot 1 = (2x+3)^{-1/2} \).
  • Multiplying these results gives us the derivative for the term \(-3 \csc(\sqrt{2x+3}) \).
By mastering the chain rule, students can handle more advanced functions with nested layers.
Trigonometric Derivatives
Trigonometric derivatives allow us to find the rates of change for trigonometric functions like sine, cosine, and cosecant. When differentiating functions involving these trigonometric terms, having a good grasp of their derivatives is vital.
  • The derivative of \( \sin(x) \) is \( \cos(x) \).
  • For \( \cos(x) \), it is \(-\sin(x) \).
  • Specifically, for the cosecant function, \( \csc(x) \), the derivative is \(-\csc(x) \cot(x) \).
In our function \(-3 \csc(\sqrt{2x+3}) \), the derivative of \(- \csc(u) \) is \(- \csc(u) \cot(u)\). We use this result combined with the chain rule to address the composite nature of the function \( -3 \csc(\sqrt{2x+3}) \):
  • Calculate these derivatives using their known differentiation rules.
  • Combine the results appropriately; for example, multiplying by any constants involved.
Getting comfortable with these trigonometric derivatives helps tackle complex calculus problems swiftly and accurately.

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Most popular questions from this chapter

Solve the given problems by finding the appropriate derivative. A connecting rod \(4 \mathrm{cm}\) long connects a piston to a crank \(2 \mathrm{cm}\) in radius (see Fig. 27.44 ). The acceleration of the piston is given by \(a=2 w^{2}(\cos \theta+0.5 \cos 2 \theta) .\) If the angular velocity \(w\) is constant, find \(\theta\) for maximum or minimum \(a\).

Solve the given problems. An object is oscillating vertically on the end of a spring such that its displacement \(d(\text { in } \mathrm{cm} \text { ) is } d=2.5 \cos 16 t, \text { where } t\) is the time (in s). What is the acceleration of the object after \(1.5 \mathrm{s} ?\)

Find the derivatives of the given functions. $$y=\sin ^{3} x-\cos 2 x$$

Solve the given problems. $$\text { Show that } \frac{d\left(\cot ^{-1} u\right)}{d x}=-\frac{1}{1+u^{2}} \frac{d u}{d x}$$

Find the differentials of the given functions. $$y=2 x \cot 3 x$$
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