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Question 1

How is this implicit equation differentiated?

x^2 +y^2 = 25

Answer 1

2x + 2y(dy/dx) = 0 since differentiating with respect to x so (dy/dx) is written

Question 2

How is the tangent line to a point on a function?

Answer 2

By first differentiating to find the derivative, and then plugging in the x and y values at the point to find slope and using point slope formula to complete

Question 3

Differentiating to find the derivative, and then plugging in the x and y values at the point to find slope and using point slope formula to complete

Answer 3

Finding the tangent line to a point on a function

Question 4

Derivative of arcsin

Answer 4

1 / √1-x^2

Question 5

1 / √1-x^2

Answer 5

Derivative of arcsin

Question 6

Derivative of arccos

Answer 6

-1 / √1-x^2

Question 7

-1 / √1-x^2

Answer 7

Derivative of arccos

Question 8

Derivative of arctan

Answer 8

1 / 1+x^2

Question 9

1 / 1+x^2

Answer 9

Derivative of arctan

Question 10

Derivative of arccot

Answer 10

-1 / 1+x^2

Question 11

-1 / 1+x^2

Answer 11

Derivative of arccot

Question 12

Derivative of arcsec

Answer 12

1 / |x|√x^2 - 1

Question 13

1 / |x|√x^2 - 1

Answer 13

Derivative of arcsec

Question 14

Derivative of arccsc

Answer 14

-1 / |x|√x^2 - 1

Question 15

-1 / |x|√x^2 - 1

Answer 15

Derivative of arccsc

Question 16

Differentiate:

xe^y = x - y

Answer 16

dy/dx = 1-e^y / e^y(x)+1

Question 17

If f(x) + x^2[f(x)]^4 = 18 and f(1)=2, find f’(1)

Answer 17

Differentiate both sides with respect to x to get -32/33

Question 18

How would you determine if two curves intersect orthogonally?

Answer 18

Take derivative of both curves and plug in the values of intersection into both derivatives. Then, set the derivatives with the values equal and if you get -1, the curves intersect orthogonally.

Question 19

How would you differentiate this equation?:

y=x^6cos(x)

Answer 19

Since a^b = e^b•ln(a), this is how you would solve it.

Question 20

Is this true or false:

If f’’(7)=0, then (7, f(7)) is an inflection point of the curve y=f(x)

Answer 20

False

Question 21

Is this true of false:

There exists a function f such that f(x)>0, f’(x)<0, and f’’(x)>0 for all x

Answer 21

True

Question 22

Is this true or false:

There exists a function f such that f(x)>0, f’(x)>0, and f’’(x)<0 for all x

Answer 22

False

Question 23

Is this true or false:

If f is even and f is differentiable, then f’ is even.

Answer 23

False

Question 24

Is this true or false:

If f is periodic and f is differentiable, then f’ is periodic

Answer 24

True

Question 25

What is the derivative of e^9x?

Answer 25

9e^9x

Question 26

What is ln(9e^9x)?

Answer 26

ln(9) + ln(e^9x)

Question 27

If f’(x)=g’(x) for 0

Answer 27

False

Question 28

What does L’Hopital’s Rule say?

Answer 28

If we have indeterminate types or 0/0 or ∞/∞ then differentiate the numerator and denominator and take the limit

Question 29

If we have indeterminate types or 0/0 or ∞/∞ then differentiate the numerator and denominator and take the limit

Answer 29

L’Hopital’s Rule

Question 30

Evaluate lim xln(x^2)

x—>0+

Answer 30

Indeterminate so use L’Hopital’s Rule to get limit = 0.

Question 31

Find the limit with either L’Hopital’s Rule or some other method.

lim (1-8x)^1/x
x—>0

Answer 31

Use the rule that a^b = e^b•ln(a) to get 1/e^8