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Chapter 3: Problem 55

In Exercises \(41-58,\) find \(d y / d t\) $$y=\tan ^{2}\left(\sin ^{3} t\right)$$

Short Answer

Expert verified
\(\frac{dy}{dt} = 6\tan(\sin^3(t)) \cdot \sec^2(\sin^3(t)) \sin^2(t)\cos(t)\).

Step by step solution

01

Identify the Outer Function

The outer function is given as the square of the tangent function, i.e., \( y = (\tan(u))^2 \). Here, the variable \( u = \sin^3(t) \).
02

Apply the Chain Rule to the Outer Function

Using the chain rule, the derivative of \((\tan(u))^2\) with respect to \(u\) is \( 2\tan(u) \cdot \sec^2(u) \cdot \frac{du}{dt} \).
03

Identify the Inner Function

The inner function is \( u = \sin^3(t) \).
04

Differentiate the Inner Function

The derivative of \(u = \sin^3(t) \) with respect to \(t\) is obtained using the chain rule again: \( \frac{du}{dt} = 3\sin^2(t)\cos(t) \).
05

Combine the Results

Substitute \( \frac{du}{dt}\) from Step 4 into the derivative expression from Step 2: \(\frac{dy}{dt} = 2\tan(\sin^3(t)) \cdot \sec^2(\sin^3(t)) \cdot 3\sin^2(t)\cos(t) \).
06

Simplify the Expression

This simplifies to: \(\frac{dy}{dt} = 6\tan(\sin^3(t)) \cdot \sec^2(\sin^3(t)) \sin^2(t)\cos(t)\).

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Key Concepts

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

Trigonometric Functions
Trigonometric functions involve angles and the relationships between angles and sides of triangles. They're fundamental in calculus and particularly important when dealing with periodic phenomena. Here are key points about the main trigonometric functions:
  • Sine ( \( \sin \) function): For a given angle in a right triangle, the sine is the ratio of the length of the side opposite the angle to the hypotenuse.

  • Cosine ( \( \cos \) function): The cosine is the ratio of the length of the adjacent side to the hypotenuse.

  • Tangent ( \( \tan \) function): The tangent of an angle is the ratio of the sine to the cosine, or the opposite side over the adjacent side.

For differentiation, focus primarily on the derivative rules for these functions:
  • The derivative of the sine function, \( \frac{d}{dt}\sin(t) = \cos(t) \) , captures the rate of change of sine with respect to the angle.

  • For cosine, it is \( \frac{d}{dt}\cos(t) = -\sin(t) \) , indicating that while one function increases with the angle, the other decreases accordingly.

  • The derivative of the tangent function is \( \frac{d}{dt}\tan(t) = \sec^2(t) \) , which describes how rapidly the tangent function changes.

Derivative of Composite Functions
A composite function combines two or more functions where the output of one function becomes the input of another. These often appear in calculus problems and need special rules for differentiation.To differentiate composite functions, the chain rule is a crucial technique. It allows us to find the derivative of complex functions formed by simpler ones:
  • Identify the "outside" function and the "inside" function. For instance, in \( y = \tan^2(\sin^3(t)) \), the outer function is \( \tan^2(u) \) and the inner function is \( \sin^3(t) \).

  • The chain rule states that the derivative of a composite function is the derivative of the outside function evaluated at the inside function times the derivative of the inside function. Symbolically, \( \frac{dy}{dt} = \frac{d}{du}(f(g(t))) \cdot \frac{dg}{dt} \).

Applying the chain rule allows handling complex derivatives more manageable by breaking them into simpler ones.
Differentiation Techniques
Differentiation techniques are crucial methods in calculus used to find the rate at which a function changes. Understanding these can greatly simplify solving problems involving derivatives, especially with trigonometric or composite functions.Some core techniques include:
  • The Chain Rule: This rule allows us to differentiate composite functions by working from the outermost function inward, as previously described. It's key when functions are nested within others, like \( y = \tan^2(\sin^3(t)) \).

  • Product Rule: Useful when differentiating the product of two functions. It states that \( (uv)' = u'v + uv' \).

  • Quotient Rule: Applied when differentiating a quotient of two functions: \( (\frac{u}{v})' = \frac{u'v - uv'}{v^2} \).

Additionally, understanding simplification and proper substitution can significantly aid in manipulating expressions to arrive at the most reduced form of a derivative, ensuring clarity and simplicity in your final answer.

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Most popular questions from this chapter

The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is $$A(q)=\frac{k m}{q}+c m+\frac{h q}{2}$$ where \(q\) is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); \(k\) is the cost of placing an order (the same, no matter how often you order); \(c\) is the cost of one item (a constant); \(m\) is the number of items sold each week (a constant) ;and \(h\) is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security. Find \(d A / d q\) and \(d^{2} A / d q^{2}\) .

Chain Rule Suppose that \(f(x)=x^{2}\) and \(g(x)=|x| .\) Then the composites $$(f \circ g)(x)=|x|^{2}=x^{2} \quad\( and \)\quad(g \circ f)(x)=\left|x^{2}\right|=x^{2}$$ are both differentiable at \(x=0\) even though \(g\) itself is not differentiable at \(x=0 .\) Does this contradict the Chain Rule? Explain.

If \(L=\sqrt{x^{2}+y^{2}}, d x / d t=-1,\) and \(d y / d t=3,\) find \(d L / d t\) when \(x=5\) and \(y=12 .\)

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365 -day year. The equation that approximates the temperature on day \(x\) is $$y=37 \sin \left[\frac{2 \pi}{365}(x-101)\right]+25$$ and is graphed in the accompanying figure. a. On what day is the temperature increasing the fastest? b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?

Tolerance The height and radius of a right circular cylinder are equal, so the cylinder's volume is \(V=\pi h^{3} .\) The volume is to be calculated with an error of no more than 1\(\%\) of the true Find approximately the greatest error that can be tolerated in the measurement of \(h\) , expressed as a percentage of \(h .\)
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