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Math
Calculus
Chapter 12
Q. Thinking back 1

Chapter 12: Q. Thinking back 1 (page 963)

Chain rule: If $f$ is a function of $x$ and $x$ is a function of $t$ , how is the chain rule used to find the rate of change of $f$ with respect to $t$ ?

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Most popular questions from this chapter

In Exercises $37 - 42$ , use the partial derivatives of $g (x, y) = \frac{\ln (y^{2} + 1)}{x}$ and the point $(3, 0)$ specified to

$(a)$ find the equation of the line tangent to the surface defined by the function in the $x$ direction,

$(b)$ find the equation of the line tangent to the surface defined by the function in the $y$ direction, and

$(c)$ find the equation of the plane containing the lines you found in parts $(a)$ and $(b)$ .

In Exercises $21 - 28$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given unit vector $u$ .

$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

$\frac{d z}{d t} w h e n z = x^{3} e^{y}, x = \sin t, a n d y = \cos t$

Describe the meanings of each of the following mathematical expressions :

$鈭� A$

Fill in the blanks to complete the limit rules. You may assume that $lim_{x 鈫� a} 鈥� f (x)$ and $lim_{x 鈫� a} 鈥� g (x)$ exists and that k is a scalar.

$lim_{x 鈫� a} 鈥� k f (x) =$

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Chain rule: If f is a function of x and x is a function of t, how is the chain rule used to find the rate of change of f with respect to t ?

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Chain rule: If $f$ is a function of $x$ and $x$ is a function of $t$ , how is the chain rule used to find the rate of change of $f$ with respect to $t$ ?