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Chapter 2: Q. 41 (page 210)

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.
$f (x) = {(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1)$

Short Answer

Expert verified

The required answer is $7 {(5 x^{4} - 3 x^{2})}^{6} (20 x^{3} - 6 x) (2 x^{3} + 1) + 6 x^{2} {(5 x^{4} - 3 x^{2})}^{7}$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = {(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1)$

02

Step 2. Calculation

Differentiate both the sides with respect tox, we get,

$f' (x) = (\frac{d}{d x} {(5 x^{4} - 3 x^{2})}^{7}) (2 x^{3} + 1) + {(5 x^{4} - 3 x^{2})}^{7} (\frac{d}{d x} (2 x^{3} + 1)) = (7 {(5 x^{4} - 3 x^{2})}^{6} (\frac{d}{d x} 5 x^{4} - \frac{d}{d x} 3 x^{2})) (2 x^{3} + 1) + 6 x^{2} {(5 x^{4} - 3 x^{2})}^{7} = 7 {(5 x^{4} - 3 x^{2})}^{6} (20 x^{3} - 6 x) (2 x^{3} + 1) + 6 x^{2} {(5 x^{4} - 3 x^{2})}^{7}$

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Most popular questions from this chapter

Velocity $v (t)$ is the derivative of position $s (t)$ . It is also true that acceleration $a (t)$ (the rate of change of velocity) is the derivative of velocity. If a race car鈥檚 position in miles t hours after the start of a race is given by the function $s (t)$ , what are the units of $s (1.2)$ ? What are the units and real-world interpretation of $v (1.2)$ ? What are the units and real-world interpretations of $a (1.2)$ ?

Use (a) the $h 鈫� 0$ definition of the derivative and then

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

29. $f (x) = x^{\frac{1}{2}}, x = 9$

Suppose f is ant cubic polynomial function $f (x) = a x^{3} + b x^{2} + c x + d$ prove that coefficients of f a, b, c, d can be expressed in terms of values of f(x) and its derivatives at the point x=2

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

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.

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

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