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Chapter 3: Problem 70

Find \(\frac{d^{2} y}{d x^{2}}\) for the following functions. $$y=\sqrt{x^{2}+2}$$

Short Answer

Expert verified
Question: Find the second derivative of the function y = 鈭�(x^2+2). Answer: The second derivative of the function is given by: $$ \frac{d^{2}y}{dx^{2}} = \frac{2}{(x^2+2)\sqrt{x^2+2}} $$

Step by step solution

01

Find the first derivative dy/dx

Differentiate the given function y with respect to x using the chain rule: $$ \frac{dy}{dx}=\frac{1}{2\sqrt{x^{2}+2}}\cdot\frac{d\left(x^{2}+2\right)}{dx} $$ Now, differentiate the inner function with respect to x: $$ \frac{d\left(x^{2}+2\right)}{dx}=2x $$ So, the first derivative becomes: $$ \frac{dy}{dx}=\frac{1}{2\sqrt{x^{2}+2}}\cdot\left(2x\right) $$
02

Simplify the first derivative

Simplify the expression for the first derivative: $$ \frac{dy}{dx}=\frac{x}{\sqrt{x^{2}+2}} $$
03

Compute the second derivative d虏y/dx虏

Differentiate the first derivative with respect to x using the quotient rule: $$ \frac{d^{2}y}{dx^{2}}=\frac{\left(\frac{d}{dx}x\right)\left(\sqrt{x^{2}+2}\right)-\left(\frac{d}{dx}\sqrt{x^{2}+2}\right)(x)}{\left(x^2+2\right)} $$
04

Compute the derivatives in the numerator

Differentiate x and sqrt(x^2+2) with respect to x: $$ \frac{d}{dx}x=1 $$ and $$ \frac{d}{dx}\sqrt{x^{2}+2}=\frac{x}{\sqrt{x^{2}+2}} $$ Substitute the derivatives into the expression for the second derivative: $$ \frac{d^{2}y}{dx^{2}}=\frac{\left(\sqrt{x^{2}+2}\right)-\left(\frac{x}{\sqrt{x^{2}+2}}\right)(x)}{(x^2+2)} $$
05

Simplify the second derivative

Simplify the expression for the second derivative: $$ \frac{d^{2}y}{dx^{2}}=\frac{x^2+2-x^2}{(x^2+2)\sqrt{x^2+2}} $$ Thus, the second derivative of the given function is: $$ \frac{d^{2} y}{d x^{2}}=\frac{2}{(x^2+2)\sqrt{x^2+2}} $$

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Most popular questions from this chapter

A hot-air balloon is \(150 \mathrm{ft}\) above the ground when a motorcycle passes directly beneath it (traveling in a straight line on a horizontal road) going \(40 \mathrm{mi} / \mathrm{hr}(58.67 \mathrm{ft} / \mathrm{s})\) If the balloon is rising vertically at a rate of \(10 \mathrm{ft} / \mathrm{s},\) what is the rate of change of the distance between the motorcycle and the balloon 10 seconds later?

One of the Leibniz Rules One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let \((f g)^{(n)}\) denote the \(n\) th derivative of the product \(f g,\) for \(n \geq 1\) a. Prove that \((f g)^{(2)}=f^{\prime \prime} g+2 f^{\prime} g^{\prime}+f g^{\prime \prime}\) b. Prove that, in general,$$(f g)^{(n)}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\\k\end{array}\right) f^{(k)} g^{(n-k)}$$ where \(\left(\begin{array}{l}n \\\ k\end{array}\right)=\frac{n !}{k !(n-k) !}\) are the binomial coefficients. c. Compare the result of (b) to the expansion of \((a+b)^{n}\).

Use any method to evaluate the derivative of the following functions. \(y=\frac{x-a}{\sqrt{x}-\sqrt{a}} ; a\) is a positive constant.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The derivative \(\frac{d}{d x}\left(e^{5}\right)\) equals \(5 \cdot e^{4}\) b. The Quotient Rule must be used to evaluate \(\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right)\) c. \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}\) d. \(\frac{d^{n}}{d x^{n}}\left(e^{3 x}\right)=3^{n} \cdot e^{3 x},\) for any integer \(n \geq 1\)

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. \(x+y^{3}-x y=1\) (Hint: Rewrite as \(y^{3}-1=x y-x\) and then factor both sides.)
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