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Chapter 2: Q. 55 (page 210)

Suppose that r is an independent variable, s is a function of r, and q is a constant. Calculate the derivatives in Exercises 53– 58. Your answers may involve r, s, q, or their derivatives.
$\frac{d}{d r} q^{3}$

Short Answer

Expert verified

$\frac{d}{d r} q^{3} = 3 q^{2} \frac{d r}{d q}$

Step by step solution

Step 1. Given Information

We need to find $\frac{d}{d r} q^{3}$

Step 2. Finding Derivative

$\frac{d}{d r} q^{3} = 3 q^{2} \frac{d r}{d q}$

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Most popular questions from this chapter

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

$f (x) = \frac{1}{x^{2} - 1}$

The following reciprocal rules tells us hoe to differentiate the reciprocal of a function

$\frac{d}{d x} (\frac{1}{f (x)}) = - \frac{1}{{[f (x)]}^{2}}$

Prove this using

a) definition of the derivative

b) by using the quotient rule

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

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A bowling ball dropped from a height of $400$ feet will be $s (t) = 400 - 16 t^{2}$ feet from the ground after $t$ seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

When the bowling ball is first dropped, with $h = 0.5,$ $h = 0.25, a n d h = 0.1$

Taking the limit: We have seen that if f is a smooth function, then $f^{'} (c) \approx \frac{f (c + h) - f (c)}{h}$ This approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

$f^{'} (c) = \lim_{h \to 0} \frac{f (c + h) - f (c)}{h}$ .

Use the limit just defined to calculate the exact slope of the tangent line to $f (x) = x^{2}$ at $x = 4 .$

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Suppose that r is an independent variable, s is a function of r, and q is a constant. Calculate the derivatives in Exercises 53– 58. Your answers may involve r, s, q, or their derivatives.ddrq3

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding Derivative

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Suppose that r is an independent variable, s is a function of r, and q is a constant. Calculate the derivatives in Exercises 53– 58. Your answers may involve r, s, q, or their derivatives.
$\frac{d}{d r} q^{3}$