Q: Find $$\frac{dy}{dx}$$ for $$e^{xy}$$ = \sec(x) + $$y^3.$$

$$\frac{e$$^{xy}$$ y + \tan(x)}{e$$^{xy}$$ x - 3y^2$}$$

Question 1

Given the curve defined by $$x^2$$ + $$2y^2 - 2xy + 3x - 3y = 5$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

Accepted Answer

$$\frac{2x - 2y + 3}{4y - 2x + 3}$$

Question 2

For the curve $$x^2$$ + $$2y^2 - 2xy + 3x - 3y = 5$, at which points is the tangent line horizontal?

Accepted Answer

Where $$2x - 2y + 3 = 0$$

Question 3

Given $$x^2 + x y$$ + $$y^2 = 7$, what is the value of $$\frac{$$d^2y$$}{$$dx^2$$}$$ at the point (3$$, -2)$$?

Accepted Answer

$$\frac{7}{27}$$

Question 4

Find $$\frac{dy}{dx}$$ for $$e^{xy}$$ = \sec(x) + $$y^3.$$

Accepted Answer

$$\frac{e$$^{xy}$$ y + 	an(x)}{e$$^{xy}$$ x - 3y^2$}$$

Question 5

Given $$y = \left(\frac{x^2$}{x^3$ + 1}\right) \sqrt{x}$$, which rule(s) must be used to find $$\frac{dy}{dx}$$?

Accepted Answer

Product rule, quotient rule, and chain rule

Question 6

Suppose $$f$$ and $$g$$ are differentiable and $$h(x) = \sqrt{f(x^2$ + 1) + 4}. If$$ $$f(5) = 3$$ and $$f'(5) = 2$$, what is $$h'(2)$$?

Accepted Answer

$$\frac{1}{2\sqrt{f(5) + 4}} \cdot f'(5) \cdot 4$$

Question 7

Let $$f(x) = \frac{3x + 1}{2x - 1}. $$What is $$(f^{-1}$$)'(2)$$?

Accepted Answer

$$\frac{4}{5}$$

Question 8

Suppose $$f(3) = 4$$, $$f'(3) = 5$$, $$f(4) = 6$$, and $$f'(4) = 7. $$Let $$g(x) = f$$^{-1}$$(x)$$ and $$h(x) = (g(x))^2$. What is h$$'(4)$$?

Accepted Answer

$$\frac{2g(4)}{f'(g(4))}$$

Question 9

Find the general antiderivative of $$f(x) = \sec^2$(x) e$$^{	an(x)}$$.

Accepted Answer

$$e^{	an(x)} + C$

Question 10

Given $$f(x) = x \cdot 5$$^{x^2}$$, what is $$\int f(x) dx$$?

Accepted Answer

$$\frac{5$$^{x^2}$$}{2 \ln 5} + C$$

Question 11

Find the general antiderivative of $$f(x) = \frac{x^2$}{(1 - 2x^3$)^2$$}$.

Accepted Answer

$$-\frac{1}{6(1 - 2x^3$)} + C$$

Question 12

A chemical reaction's rate is modeled by $$y = \frac{(3x + 2)^4$}{(x - 1)^2$ \sqrt{4x + 5}}. $$Which differentiation technique is most efficient to find $$\frac{dy}{dx}$$?

Accepted Answer

Logarithmic differentiation

Question 13

Given $$\ln(xy) = e$$^{x^2 + y^2}$$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

Accepted Answer

$$\frac{e$$^{x^2 + y^2}$$ - \frac{1}{x}}{\frac{1}{y} + 2y e$$^{x^2 + y^2}$$}$$

Question 14

Suppose $$h(x) = |f(x) - 2| + \frac{(g(x))^2$}{3}. $$Which rule(s) must be used to find $$h'(x)$$?

Accepted Answer

Chain rule, absolute value derivative, and power rule

Question 15

Given $$h(x) = \arccot(f(x)) - f(x)g'(x)$$, which of the following is the correct expression for $$h'(x)$$?

Accepted Answer

$$-\frac{f'(x)}{1 + (f(x))^2$} - f'(x)g'(x) - f(x)g''(x)$$

Question 16

A population grows according to $$y = (g(x^2$ + x))^2$. If g(2) = 3$ and g$$'(2) = 4$$, what is $$\frac{dy}{dx}$$ at x = -2$?

Accepted Answer

$$2g(2)g'(2)(2(-2) + 1)$$

Question 17

Find the general antiderivative of $$f(x) = \frac{e$$^{2x}$$}{e$$^{2x}$$ - 1}.$$

Accepted Answer

$$\frac{1}{2} \ln|e$$^{2x}$$ - 1| + C$$

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