Ximpledu

Derivative Function

See how to find the derivative function
(logarithm, exponential, trigonometry).
examples and their solutions.

Derivative of ln x

Formula

[ln x]' = 1x
[ln |x|]' = 1x

Example

y = ln (2x + 7)

y' = ?
Solution

Example

y = ln |x3 - 2x|

y' = ?
Solution

Example

y = ln x4

y' = ?
Solution

Example

y = x3 ln x

y' = ?
Solution

Example

y = xx

y' = ?
Solution

Derivative of loga x

Formula

[loga x]' = 1x ln a
[loga |x|]' = 1x ln a

Example

y = log2 (x3 - 8x)

y' = ?
Solution

Example

y = log5 5x6

y' = ?
Solution

Derivative of ex

Formula

[ex]' = ex

Example

y = x2ex

y' = ?
Solution

Example

y = ex2 + 4x

y' = ?
Solution

Derivative of ax

Formula

[ax]' = ax ln a

Example

y = 22x + 1

y' = ?
Solution

Derivative of sin x

Formula

[sin x]' = cos x

Example

y = x sin x

y' = ?
Solution

Example

y = sin x2

y' = ?
Solution

Example

y = sin2 x

y' = ?
Solution

Derivative of cos x

Formula

[cos x]' = -sin x

Example

y = cos (x3 - 4)

y' = ?
Solution

Derivative of tan x

Formula

[tan x]' = sec2 x

Example

y = x2 tan x

y' = ?
Solution

Derivative of csc x

Formula

[csc x]' = -csc x cot x

Example

y = csc (1 - x2)

y' = ?
Solution

Derivative of sec x

Formula

[sec x]' = sec x tan x

Example

y = sec3 x

y' = ?
Solution

Derivative of cot x

Formula

[cot x]' = -csc2 x

Example

y = 7x cot x

y' = ?
Solution

sinh x, cosh x: Definition

Definition

sinh x = ex - e-x2
cosh x = ex - e-x2
(cos x, sin x) is on the unit circle x2 + y2 = 1.
(ex - e-x2, ex - e-x2) is on the hyperbola
x2 - y2 = 1.
So people made sinh x and cosh x.
(hyperbolic sine, hyperbolic cosine)
→ (cosh x, sinh x) is on the hyperbola x2 - y2 = 1.

Derivative of sinh x

Formula

[sinh x]' = cosh x

Example

y = sinh x5

y' = ?
Solution

Derivative of cosh x

Formula

[cosh x]' = sinh x

Example

y = cosh2 x

y' = ?
Solution