Q 1. TB Using the chain rule: Given that... [FREE SOLUTION]

Chapter 3: Q 1. TB (page 298)

Using the chain rule: Given that $r = r (t), s = s (t),$ and $u = u (t)$ are functions of $t$ and that $c$ and $k$ are constants, find each of the following derivatives.
role="math" localid="1649774108945" $(i) \frac{d}{d t} (蟺 u^{2}) (i i) \frac{d}{d t} (3 r + 2 s) (i i i) \frac{d}{d t} (c u + r s) (i v) \frac{d}{d t} (k + u^{3}) (v) \frac{d}{d t} (c r^{2} u) (v i) \frac{d}{d t} (\frac{c + s}{k + u})$

Short Answer

Expert verified

The derivative of $(i) \frac{d}{d t} (蟺 u^{2}) = 2 蟺 u [u^{'} (t)] (i i) \frac{d}{d t} (3 r + 2 s) = 3 [r^{'} (t)] + 2 [s^{'} (t)] (i i i) \frac{d}{d t} (c u + r s) = c [u^{'} (t)] + r [s^{'} (t)] + s [r^{'} (t)] (i v) \frac{d}{d t} (k + u^{3}) = 3 u^{2} [u^{'} (t)] (v) \frac{d}{d t} (c r^{2} u) = c \{r^{2} [u^{'} (t)] + 2 u r [r^{'} (t)]\} (v i) \frac{d}{d t} (\frac{c + s}{k + u}) = \frac{s^{'} (t) (k + u) - u^{'} (t) (c + s)}{{(k + u)}^{2}}$

Step by step solution

Step 1. Definition

In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions $f$ and $g$ in terms of the derivatives of $f$ and $g$ .

Step 2. Explanation and solution

$(i) \frac{d}{d t} (蟺 u^{2}) = 蟺 \frac{d}{d t} (u^{2}) = 2 蟺 u [u^{'} (t)]$

$(i i) \frac{d}{d t} (3 r + 2 s) = 3 \frac{d}{d t} (r) + 2 \frac{d}{d t} (s) = 3 [r^{'} (t)] + 2 [s^{'} (t)]$

$(i i i) \frac{d}{d t} (c u + r s) = c \frac{d}{d t} (u) + r \frac{d}{d t} (s) + s \frac{d}{d t} (r) = c [u^{'} (t)] + r [s^{'} (t)] + s [r^{'} (t)]$

$(i v) \frac{d}{d t} (k + u^{3}) = \frac{d}{d t} (k) + \frac{d}{d t} (u^{3}) = 3 u^{2} [u^{'} (t)]$

$(v) \frac{d}{d t} (c r^{2} u) = c [r^{2} \frac{d}{d t} (u) + u \frac{d}{d t} (r^{2})] = c \{r^{2} [u^{'} (t)] + 2 u r [r^{'} (t)]\}$

$(v i) \frac{d}{d t} (\frac{c + s}{k + u}) = \frac{(k + u) \frac{d}{d t} (c + s) - (c + s) \frac{d}{d t} (k + u)}{{(k + u)}^{2}} = \frac{s^{'} (t) (k + u) - u^{'} (t) (c + s)}{{(k + u)}^{2}}$

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Using the chain rule: Given that r=rt,s=st,and u=utare functions of tand that cand kare constants, find each of the following derivatives. role="math" localid="1649774108945" iddt蟺u2iiddt3r+2siiiddtcu+rsivddtk+u3vddtcr2uviddtc+sk+u

Short Answer

Step by step solution

Step 1. Definition

Step 2. Explanation and solution

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Applied Mathematics

Statistics

Logic and Functions

Calculus

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