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Chapter 2: Q 33. (page 222)

Find the derivatives of the functions: $f (x) = x^{- 3} e^{2 x}$

Short Answer

Expert verified

The required answer is $2 x^{- 3} e^{2 x} - 3 x^{- 4} e^{2 x}$ .

Step by step solution

Step 1. Given information.

The given function is: $f (x) = x^{- 3} e^{2 x}$

Step 2. Find the derivative of the given function.

$f (x) = x^{- 3} e^{2 x} f' (x) = \frac{d}{d x} (x^{- 3} e^{2 x}) = x^{- 3} \frac{d}{d x} e^{2 x} + e^{2 x} \frac{d}{d x} x^{- 3} = 2 x^{- 3} e^{2 x} - 3 x^{- 4} e^{2 x}$

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Most popular questions from this chapter

For each function $f (x)$ and interval localid="1648297458718" $[a, b]$ in Exercises localid="1648297462718" $81 - 86$ , use the Intermediate Value Theorem to argue that the function must have at least one real root on localid="1648297466951" $[a, b]$ . Then apply Newton’s method to approximate that root.

localid="1648297471865" $f (x) = x^{3} - 3 x + 1, [a, b] = [1, 2]$

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 34-59

$f (x) = 3 \sqrt{x}$

Write down a rule for differentiating a composition $f (u (v (w (x))))$ of four functions

(a) in “prime” notation and

(b) in Leibniz notation.

Use (a) the $h \to 0$ definition of the derivative and then

(b) the $z \to c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23–38.

28. $f (x) = x^{4} + 1, x = 2$

In Exercises 69–80, determine whether or not $f$ is continuous and/or differentiable at the given value of $x$ . If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

$f (x) = \{\begin{cases} x^{2}, i f x \leq 1 \\ 2 x - 1, i f x > 1, \end{cases} x = 1$

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Find the derivatives of the functions:f(x)=x-3e2x

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the given function.

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Most popular questions from this chapter

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Find the derivatives of the functions: $f (x) = x^{- 3} e^{2 x}$