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Chapter 27: Problem 12

Find the derivatives of the given functions. $$y=3 x^{4}+2 \tan ^{2}\left(x^{2}\right)$$

Short Answer

Expert verified
The derivative is \(12x^3 + 8x\tan(x^2)\sec^2(x^2)\).

Step by step solution

01

Understand the Function Components

The given function is \(y = 3x^4 + 2\tan^2(x^2)\). This function has two parts: \(3x^4\) and \(2\tan^2(x^2)\). We will need to find the derivative of each part separately and then sum them up.
02

Differentiate the First Term

Differentiate \(3x^4\) using the power rule: \(\frac{d}{dx}[ax^n] = nax^{n-1}\). Apply the rule: \(\frac{d}{dx}[3x^4] = 4 \cdot 3x^{4-1} = 12x^3\).
03

Differentiate the Second Term

The second term is \(2\tan^2(x^2)\). Use the chain rule to differentiate: \(u = x^2\), then \(\tan^2(u)\) is \([\tan(u)]^2\). First differentiate the outer function: \(\frac{d}{du} [\tan^2(u)] = 2\tan(u) \cdot \sec^2(u)\). Next, differentiate the inner function \(x^2\): \(\frac{d}{dx}[x^2] = 2x\). Apply the chain rule to get \(\frac{d}{dx}[\tan^2(x^2)] = 2\tan(x^2)\sec^2(x^2) \cdot 2x = 4x\tan(x^2)\sec^2(x^2)\). Multiply by 2: \(8x\tan(x^2)\sec^2(x^2)\).
04

Combine the Derivatives

Now combine the derivatives from Steps 2 and 3. The derivative of \(y = 3x^4 + 2\tan^2(x^2)\) is \(12x^3 + 8x\tan(x^2)\sec^2(x^2)\).

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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 simple and vital tool in calculus for finding derivatives. It is commonly used to differentiate polynomial terms of the form \(ax^n\). The power rule states:
  • Take the exponent \(n\) and bring it down as a coefficient.
  • Reduce the exponent by one, meaning your new exponent becomes \(n-1\).
For example, if you want to differentiate \(3x^4\), apply the power rule:- Bring down the 4: it becomes \(4 \times 3\).- Reduce the exponent 4 by one to get 3.- Thus, the derivative is \(12x^3\).
Use this rule every time you have a simple power of \(x\) to simplify your differentiation process.
Chain Rule
The Chain Rule is essential when dealing with composite functions. Think of it as a method to "unroll" a complicated function into simpler parts that you can differentiate. When you have a function inside another function, like \(\tan^2(x^2)\), the chain rule comes into play.
  • First, identify the 'outer' function; for \(\tan^2(x^2)\), it is \([\tan(x)]^2\).
  • Differentiate the outer function ignoring the inner part: \(2\tan(x)\sec^2(x)\).
  • Next, differentiate the 'inner' function \(x^2\), which gives \(2x\).
  • Finally, multiply these derivatives together.
Apply this systematically, and it breaks down complex differentiation into manageable chunks. In our exercise, it simplifies \(2\tan^2(x^2)\) to \(8x\tan(x^2)\sec^2(x^2)\).
Trigonometric Derivatives
Trigonometric derivatives help differentiate functions that involve sine, cosine, tangent, etc. Here, knowing specific derivatives like those of \(\tan(x)\) is crucial.
  • For \(\tan(x)\), the derivative is \(\sec^2(x)\).
  • You often deal with composite functions, like \(\tan^2(x)\), requiring the application of both trigonometric derivatives and the chain rule.

In the provided exercise, understanding \(\tan^2(x^2)\) uses this knowledge. The derivative of \(\tan(x^2)\) uses the rule \(\sec^2(x)\) combined with the chain rule, producing an additional \(x\) factor from the inner function differentiation.
Grasp these trigonometric derivatives and master the chain rule simultaneously for precision in solving complex derivatives.

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Most popular questions from this chapter

Solve the given problems by finding the appropriate derivative. A package of weather instruments is propelled into the air to an altitude of about \(7 \mathrm{km} .\) A parachute then opens, and the package returns to the surface. The altitude \(y\) of the package as a function of the time \(t\) (in \(\min\) ) is given by \(y=\frac{10 t}{e^{0.4 t}+1} \cdot\) Find the vertical velocity of the package for \(t=8.0\) min.

Solve the given problems by finding the appropriate derivative. The power supply \(P(\text { in } W\) ) in a satellite is \(P=100 e^{-0.005 t}\), where \(t\) is measured in days. Find the time rate of change of power after 100 days.

Evaluate each limit (if it exists). Use \(L\) Hospital's rule (if appropriate). $$\lim _{x \rightarrow 1} \frac{\sin \pi x}{x-1}$$

Evaluate each limit (if it exists). Use \(L\) Hospital's rule (if appropriate). $$\lim _{\theta \rightarrow \pi / 2} \frac{1+\sec \theta}{\tan \theta}$$

Evaluate each limit (if it exists). Use \(L\) Hospital's rule (if appropriate). $$\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}-2}{1-\cos 2 x}$$
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