Standard Derivatives

Question 1

d/dx (e^x) = ?

where e is the exponential function

Answer 1

e^x

Question 2

d/dx (e^ax) = ?

Answer 2

ae^ax

Question 3

What is the general rule (in words) when differentiating exponential funtions?

Answer 3

To differentiate an exponential funtion, you differentiate the power and then multiply that differential by the original function.

Question 4

d/dx (ln(ax^2)) = ?

Answer 4

2ax/(ax^2)

Question 5

d/dx (ln(x)) = ?

where ln is the natural logarithm and** x > 0**

Answer 5

1/x

Question 6

What is the general rule (in words) when differentiating natural logarithms?

Answer 6

To differentiate a natural log funtion, you differentiate the inner function and divide that by the inner function

Question 7

What is the general rule (expressed mathematically) when differentiating natural logarithms?

Answer 7

d/dx (ln(ax)) = a/ax

Question 8

d/dx (logp(x)) = ?

the ‘a’ here is logarithm to base a

Answer 8

1/x * ln (p)

this is the same thing as with natural log, which ought to be 1/x multiplied by ln(e) but ln(e) = 1

Question 9

What is the general rule (in words) when differentiating logarithms?

be it natural or normal

Answer 9

To differentiate a log funtion to a given base, p, you differentiate the inner function and divide that by the inner function and multiply the whole thing by the natural log of the base, p.

If p=e in the case of ln, ln(e) = 1

Question 10

d/dx (a^x) = ?

where a > 0

Answer 10

(a^x) * ln(a)

Question 11

d/dx (sin(x)) = ?

Answer 11

cos(x)

remember, sine does not have trouble

Question 12

d/dx (cos(x)) = ?

Answer 12

-sin(x)

cosine must have comma

Question 13

How can you get d/dx (tan(x))

Answer 13

using the quotient rule and the idea that tan(x) = sin(x)/cos(x)

Question 14

How can you get d/dx (sec(x))

Answer 14

using the quotient rule and the idea that sec(x) = 1/cos(x)

Question 15

How can you get d/dx (csc(x))

Answer 15

using the quotient rule and the idea that csc(x) = 1/sin(x)

Question 16

How can you get d/dx (cot(x))

Answer 16

using the quotient rule and the idea that cot(x) = 1/tan(x) = cos(x)/sin(x)