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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

Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.

$\frac{\partial z}{\partial s} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

$f (x, y) = \frac{x + y}{x^{2} + y^{2}}, P = (1, 2), v = ⟨ 3, - 2 ⟩$

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 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.

$\frac{\partial z}{\partial θ} w h e n z = (x^{2} + x y) e^{y}, x = r \cos θ a n d y = r \sin θ$

In Exercises $37 - 42$ , use the partial derivatives of role="math" localid="1650186824938" $f (x, y) = x^{y}$ and the point $(e, 3)$ 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 $(b)$ .

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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 ?

Short Answer

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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 $t$ ?