Question

Find the first derivatives of (a) \(x^{2} \exp x\), (b) \(2 \sin x \cos x\), (c) \(\sin 2 x,(d) x \sin a x\), (e) \((\exp a x)(\sin a x) \tan ^{-1} a x\), (f) \(\ln \left(x^{2}+x^{-a}\right)\), (g) \(\ln \left(a^{x}+a^{-x}\right)\), (h) \(x^{x}\).

Short answer

(a) \(2x + x^2\) \exp x, (b) 2 \cos 2x, (c) 2 \cos 2x, (d) \sin ax + ax \cos ax, (e) use product rule multiple times, complex, (f) \(2x - ax^{-a-1}\)/\(x^2 + x^{-a}\), (g) \(a^x \ln a - a^{-x} \ln a\)/\(a^x + a^{-x}\), (h) x^x (\ln x + 1).

Step-by-step solution

- Derivative of \(x^{2} \exp x\)

Use the product rule for differentiation: \((uv)' = u'v + uv'\). Here, \(u = x^2\) and \(v = \exp x\). \ u' = \(2x\), v' = \exp x. The derivative is \(u'v + uv' = 2x \exp x + x^2 \exp x\). Simplify to \((2x + x^2) \exp x\).

- Derivative of \(2 \sin x \cos x\)

Recognize that \(2 \sin x \cos x = \sin 2x\) by using the double-angle identity. The derivative of \(\sin 2x\) is \(2 \cos 2x\).

- Derivative of \(\sin 2 x\)

The derivative of \(\sin 2x\) is found using the chain rule. Let \(u = 2x\). The derivative of \(\sin u\) is \(\cos u\). Therefore, the derivative is \(\cos 2x \times 2\), resulting in \(2 \cos 2 x\).

- Derivative of \(x \sin a x\)

Use the product rule: \(u = x\), \(v = \sin ax\), \(u' = 1\), \(v' = a \cos ax\). Applying the product rule: \(u'v + uv' = 1 \sin ax + x \times a \cos ax\), the result is \(\sin ax + ax \cos ax\).

- Derivative of \((\exp a x)(\sin a x) \tan ^{-1} a x\)

This function requires the product rule applied multiple times. Consider \(u = \exp ax\), \(v = \sin ax\), and \(w = \tan^{-1} ax\). Compute each term's derivative: \(u' = a \exp ax\), \(v' = a \cos ax\), and \(w' = \frac{a}{1 + (ax)^2}\). The result requires combining these using the product rule, leading to a complex expression.

- Derivative of \(\ln\left(x^{2}+x^{-a}\right)\)

Use the chain rule: Let \(u = x^2 + x^{-a}\). The derivative of \(\ln u\) is \(\frac{1}{u} \times u'\). Now, find \(u' = 2x - ax^{-a-1}\). Combine results to get \(\frac{2x - ax^{-a-1}}{x^2 + x^{-a}}\).

- Derivative of \(\ln \left(a^{x} + a^{-x}\right)\)

Use the chain rule: Let \(u = a^x + a^{-x}\). The derivative of \(\ln u\) is \(\frac{1}{u} \times u'\). Now, find \(u' = a^x \ln a - a^{-x} \ln a\). Combine results to get \(\frac{a^x \ln a - a^{-x} \ln a}{a^x + a^{-x}}\).

- Derivative of \(x^{x}\)

Rewrite \(x^x\) as \(\exp(x \ln x)\). Use the chain rule: Let \(u = x \ln x\). The derivative is \(\exp(u) \times u'\). Now, find \(u' = \ln x + 1\). Combine results to get \(x^x \times (\ln x + 1)\).

Key concepts

product rule

The product rule is a fundamental concept in calculus for finding the derivative of a product of two functions. It states that the derivative of a product of two functions, say \(u(x)\) and \(v(x)\), can be given by:
\((uv)' = u'v + uv'\)
For example, in the problem, finding the derivative of \(x^2 \exp(x)\) involves identifying \(u = x^2\) and \(v = \, \exp(x)\).
Following the product rule:

• Find \(u' = 2x\)
• Find \(v' = \, \exp(x)\)

Combine these to get:
\(u'v + uv' = 2x \, \exp(x) + x^2 \, \exp(x)\)
Simplifying:
\((2x + x^2) \, \exp(x)\)
Using the product rule will be essential in more complex derivatives, such as those with three or more terms multiplied together.

chain rule

The chain rule is another crucial technique in differentiation when dealing with composite functions. It provides a method to differentiate a function that is nested within another function.
In formal terms, if you have a function \(y = f(g(x))\), the chain rule states:
\( \frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)
This is applied in the problem where finding the derivative of \(\sin 2x\) is required:

• Set \(u = 2x\)
• The derivative of \(\sin u\) is \(\cos u\)
• The derivative of \(2x\) is 2

Combining with the chain rule:
\( \cos 2x \cdot 2 = 2 \cos 2x\)
The chain rule comes in handy for other complex derivatives and is particularly beneficial when dealing with functions within functions.

trigonometric derivatives

Trigonometric derivatives are imperative when handling functions involving sine, cosine, and other trigonometric expressions. Here are some fundamental trigonometric derivatives to remember:

• \( \frac{d}{dx}(\sin x) = \cos x \)
• \( \frac{d}{dx}(\cos x) = -\sin x \)
• \( \frac{d}{dx}(\tan x) = \sec^2 x \)

For instance, consider the derivative of \(2 \sin x \cos x\). Utilizing the double-angle identity, \(2 \sin x \cos x = \sin 2x\):
Then, apply the chain rule:
\( \frac{d}{dx}(\sin 2x) = 2 \cos 2x\)
Understanding trigonometric derivatives can simplify working with trigonometric integrals and identities.

logarithmic differentiation

Logarithmic differentiation is a powerful tool used to simplify finding the derivative of products, quotients, or power functions. It involves taking the natural logarithm of both sides of an equation for which you want to find the derivative, and then differentiating.
For example, to find the derivative of \( \ln(x^2 + x^{-a})\):

• Let \( u = x^2 + x^{-a} \)
• The derivative of \( \ln u\) is \( \frac{1}{u} \cdot u'\)
• Then find \(u' = 2x - ax^{-a-1}\)

Combining them:
\( \frac{1}{x^2 + x^{-a}} \cdot (2x - ax^{-a-1})\)
Logarithmic differentiation proves extremely useful, especially in dealing with functions raised to other functions, like in the problem of differentiating \(x^x\):
First, rewrite \(x^x\) as \( \exp(x \ln x)\). Then, differentiate:
\( \exp(x \ln x) \cdot (\ln x + 1)\)
Overall, logarithmic differentiation is a great simplification technique.