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Chapter 2: Q 24. (page 222)

Find the derivatives of the functions: $f (x) = \frac{e^{x} \ln x}{x^{2} 鈭� 1}$

Short Answer

Expert verified

The required answer is $\frac{(x^{2} - 1) (e^{x} \frac{1}{x} + e^{x} \ln x) + 2 x e^{x} \ln x}{{(x^{2} - 1)}^{2}}$ .

Step by step solution

01

Step 1. Given information.

The given function is: $f (x) = \frac{e^{x} \ln x}{x^{2} 鈭� 1}$

02

Step 2. Find the derivative of the given function.

$f (x) = \frac{e^{x} \ln x}{x^{2} 鈭� 1} f' (x) = \frac{d}{d x} (\frac{e^{x} \ln x}{x^{2} 鈭� 1}) = \frac{(x^{2} 鈭� 1) \frac{d}{d x} (e^{x} \ln x) + e^{x} \ln x \frac{d}{d x} (x^{2} 鈭� 1)}{{(x^{2} 鈭� 1)}^{2}} = \frac{(x^{2} 鈭� 1) [e^{x} \frac{d}{d x} \ln x + \ln x \frac{d}{d x} e^{x}] + e^{x} \ln x 2 x}{{(x^{2} 鈭� 1)}^{2}} = \frac{(x^{2} 鈭� 1) [e^{x} \frac{1}{x} + e^{x} \ln x] + 2 x e^{x} \ln x}{{(x^{2} 鈭� 1)}^{2}}$

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Most popular questions from this chapter

Suppose f is a polynomial of degree n and let k be some integer with $0 鈮� k 鈮� n$ . Prove that if f(x) is of the form $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . . + a_{1} x + a_{0}$

Then $a_{k} = \frac{f^{k} (0)}{k!}$ where $f^{k} (x)$ is the k-th derivative of $f (x) a n d k! = k 路 (k - 1) 路 (k - 2) . . . . 2 路 1$

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = \frac{{(1 + \sqrt{x})}^{2}}{3 x^{2} - 4 x + 1}$

Use the $h 鈫� 0$ definition of the derivative to prove the power rule holds for positive integers powers

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 39-54

role="math" localid="1648290170541" $f (x) = \frac{x - 1}{x + 3}$

Suppose $u (x) = \sqrt{3 x^{2} + 1}$ and $f (u) = \frac{u^{2} + 3 u^{5}}{1 - u}$ . Use the chain rule to find role="math" localid="1648356625815" $\frac{d}{d x} (f (u (x)))$ without first finding the formula for $f (u (x))$ .

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