Q: A chemical reaction rate is modeled by $$R(x) = \ln\left(\sqrt{\frac{x^2$ \cos x}{1 + e$$^{2x}$$}}\right). $$What is the derivative $$R'(x)$$?

$$R'(x) = \frac{1}{x} - \tan x - \frac{e$$^{2x}$$}{1 + e$$^{2x}$$}$$

Question 1

Find $$y'(x)$$ for the function $$y = \ln(x + e^x) + 2x^7$ + 5x$$^{12}$$ + 6^x.$$

Accepted Answer

$$y'(x) = \frac{1 + e^x}{x + e^x} + 14x^6$ + 60x$$^{11}$$ + 6^x \ln 6$$

Question 2

Use implicit differentiation to find $$\frac{dy}{dx}$$ given $$\sin y = x \cos y - 2.$$

Accepted Answer

Question 3

Use properties of logarithms to rewrite $$y = \ln\left(\sqrt{\frac{x^2$ \cos x}{1 + e$$^{2x}$$}}\right)$$ and then find $$\frac{dy}{dx}.$$

Accepted Answer

Question 4

Use logarithmic differentiation to find $$\frac{dy}{dx}$$ given $$x^{\sin x}$$ = \ln y$$.

Accepted Answer

Question 5

Find the derivative of $$y = 2 \sin$$^{-1}$$(e^x + x) - 3 \cos$$^{-1}$$(9x) + 4 	an$$^{-1}$$(2x - 5).$$

Accepted Answer

$$y'(x) = \frac{2(e^x + 1)}{\sqrt{1 - (e^x + x)^2$}} - \frac{27}{\sqrt{1 - (9x)^2$}} + \frac{8}{1 + (2x - 5)^2$}$$

Question 6

Let $$x = x(t)$$ and $$y = y(t). If$$ $$x^2$$ $$y^3 = 4/27$ and $$\frac{dy}{dt} = 1/2$$, what is $$\frac{dx}{dt}$$ when x = 2$?

Accepted Answer

$$\frac{dx}{dt} = -\frac{3}{8}$$

Question 7

A biologist models the population of bacteria by $$P(t) = \ln(t + e^t) + 2t^7$ + 5t$$^{12}$$ + 6^t. $$What is the instantaneous rate of change of the population at $$t = 0$$?

Accepted Answer

$$P'(0) = 1 + 0 + 0 + \ln 6$$

Question 8

Given $$y = \ln(x + e^x)$$, which of the following is the correct expression for $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{1 + e^x}{x + e^x}$$

Question 9

Suppose $$y = 2 \sin$$^{-1}$$(e^x + x). $$What is $$\frac{dy}{dx} at$$ $$x = 0$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2}{\sqrt{1 - (e^0$ + 0)^2$}}$$

Question 10

A chemical reaction rate is modeled by $$R(x) = \ln\left(\sqrt{\frac{x^2$ \cos x}{1 + e$$^{2x}$$}}\right). $$What is the derivative $$R'(x)$$?

Accepted Answer

$$R'(x) = \frac{1}{x} - 	an x - \frac{e$$^{2x}$$}{1 + e$$^{2x}$$}$$

Question 11

Given $$x^2$$ $$y^3 = 4/27$ and $$\frac{dy}{dt} = 1/2$$, what is $$\frac{dx}{dt}$$ when x = 2$ and y = 1$?

Accepted Answer

$$\frac{dx}{dt} = -\frac{3}{8}$$

Question 12

If $$y = \ln(x + e^x)$$ models the growth of a tumor, what is the rate of change of the tumor size at $$x = 1$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{1 + e^1$}{1 + e^1$}$$

Question 13

For $$y = 2 \sin$$^{-1}$$(e^x + x) - 3 \cos$$^{-1}$$(9x) + 4 	an$$^{-1}$$(2x - 5)$$, which term contributes a negative value to $$y'(x)$$?

Accepted Answer

$$-3 \cos$$^{-1}$$(9x)$$

Question 14

Given $$x^{\sin x}$$ = \ln y$$, use logarithmic differentiation to find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{dy}{dx} = y \left( \sin x \ln x + \cos x \ln x \right)$$

Question 15

A physicist models the displacement of a particle by $$y = 2 \sin$$^{-1}$$(e^x + x). $$What is the velocity at $$x = 0$$?

Accepted Answer

$$v = \frac{2}{\sqrt{1 - (e^0$ + 0)^2$}}$$

Question 16

If $$y = \ln(x + e^x) + 2x^7$ + 5x$$^{12}$$ + 6^x$$ models the cost of production, what is the marginal cost at $$x = 1$$?

Accepted Answer

$$MC = \frac{1 + e^1$}{1 + e^1$} + 14 \cdot 1^6$ + 60 \cdot 1$$^{11}$$ + 6 \ln 6$$

Question 17

Given $$x^2$$ $$y^3 = 4/27$ and $$\frac{dy}{dt} = 1/2$$, what is $$\frac{dx}{dt}$$ when x = 2$ and y = 1$?

Accepted Answer

$$\frac{dx}{dt} = -\frac{3}{8}$$

Differentiation Techniques in Calculus: Study Notes and Applications

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