Problem 51 - Calculus Early Transcendentals

Question

Find the derivatives of the following functions. $$f(x)=x \sinh ^{-1} x-\sqrt{x^{2}+1}$$

Step-by-Step Solution

Answer

Answer: The derivative of the given function is $$f'(x) = \sinh^{-1}x$$.

Step 1: Identify the function components

The function consists of two parts: $$x \sinh^{-1}x$$ and $$-\sqrt{x^2 + 1}$$. We will find the derivatives of both parts and then combine them to find the derivative of the whole function.

Step 2: Derivative of the first part

Let's find the derivative of the first part of the function, $$x \sinh^{-1}x$$. We will use the product rule: $$(u \cdot v)' = u' \cdot v + u \cdot v'$$ Here, $$u = x$$ and $$v = \sinh^{-1}x$$. First, we need to find the derivative of $$\sinh^{-1}x$$, for which we will use the chain rule: $$\frac{d}{dx}(\sinh^{-1}x) = \frac{1}{\sqrt{x^2 + 1}}$$ Now, apply the product rule: $$\frac{d}{dx}(x \cdot \sinh^{-1}x) = \frac{d}{dx}(x) \cdot \sinh^{-1}x + x \cdot \frac{d}{dx}(\sinh^{-1}x)$$ $$= 1 \cdot \sinh^{-1}x + x \cdot \frac{1}{\sqrt{x^2+1}}$$

Step 3: Derivative of the second part

Now, we will find the derivative of the second part of the given function, $$-\sqrt{x^2 + 1}$$. Using the chain rule, we have: $$-\frac{d}{dx}\sqrt{x^2 + 1} = -\frac{1}{2}(x^2 + 1)^{-\frac{1}{2}}\cdot\frac{d}{dx}(x^2 + 1)$$ $$= -\frac{1}{2}(x^2 + 1)^{-\frac{1}{2}}\cdot(2x)$$ $$= -\frac{x}{\sqrt{x^2 + 1}}$$

Step 4: Combine both parts

Now, we combine the derivatives of both parts: $$f'(x) = \frac{d}{dx}(x \sinh^{-1}x - \sqrt{x^2 + 1})$$ $$= (\sinh^{-1}x + \frac{x}{\sqrt{x^2+1}}) - \frac{x}{\sqrt{x^2 + 1}}$$

Step 5: Simplify the expression

Finally, let's simplify the expression: $$f'(x) = \sinh^{-1}x$$ So, the derivative of the given function is $$f'(x) = \sinh^{-1}x$$.

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions offer a delightful area of calculus. They are similar to the classic trigonometric functions but applicable in hyperbolic geometry. Particularly, the inverse hyperbolic sine function, \( \sinh^{-1}(x) \), plays a central role in our exercise. It might seem complex, but this function is simply a transformation of the natural logarithm:

• \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \)

Understanding how to derive the formula for the inverse function helps in recognizing patterns and properties when solving problems.
When it comes to finding the derivative of \( \sinh^{-1}(x) \), you apply a specific rule. This rule states that:\[\frac{d}{dx} (\sinh^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}}\]This derivative is functional due to how the hyperbolic sine relates to the unit hyperbola, and it can make solving these types of derivatives simpler with practice.

Product Rule

The product rule is a fundamental tool in calculus, essential when differentiating products of two functions. This rule states that if you have two functions, say \( u(x) \) and \( v(x) \), the derivative of their product is: \[ (u \cdot v)' = u' \cdot v + u \cdot v' \]In our example function \( f(x) = x \sinh^{-1} x \), we recognize it as a product of \( u = x \) and \( v = \sinh^{-1}x \).
To differentiate it using the product rule, follow these steps:

• First, find \( u' \), which in this case, the derivative of \( x \) is 1.
• Second, find \( v' \), the derivative of \( \sinh^{-1}x \) is \( \frac{1}{\sqrt{x^2 + 1}} \).
• Finally, substitute into the product rule: \( 1 \cdot \sinh^{-1}x + x \cdot \frac{1}{\sqrt{x^2 + 1}} \).

The product rule, by simplifying these steps, allows for efficient differentiation of more intricate equations.

Chain Rule

The chain rule is an essential technique used to differentiate composite functions. It helps in breaking down complicated expressions by differentiating each layer of the function separately. In simpler terms, if you have a function \( y = g(h(x)) \), the derivative is computed by multiplying the derivative of \( g \) with respect to \( h \), \( g'(h(x)) \), and the derivative of \( h \) with respect to \( x \).This is expressed mathematically as:\[\frac{dy}{dx} = g'(h(x)) \cdot h'(x)\]In our example, it's utilized to find the derivative of the second part of the function, \(-\sqrt{x^2 + 1}\).
By recognizing \( h(x) = x^2 + 1 \), and \( g(h) = \sqrt{h} \), you use the chain rule this way:

• First, differentiate \( g(h) \) to get \( \frac{-1}{2}\cdot(x^2 + 1)^{-1/2} \).
• Then, differentiate \( h(x) = x^2 + 1 \) to obtain \( 2x \).
• Multiply these derivatives: \(-\frac{x}{\sqrt{x^2 + 1}} \).

The chain rule not only simplifies and clarifies the process but is vital in handling nested functions efficiently.