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 y' = ?
y = ln (2x + 7)
y' = 12x + 7⋅2 - [1] [2]
= 22x + 7
y' = 12x + 7⋅2 - [1] [2]
= 22x + 7
Close
Example
y = ln |x3 - 2x|
y' = ?
Solution y' = ?
y = ln |x3 - 2x|
y' = 1x3 - 2x⋅(3x2 - 2)
= 3x2 - 2x3 - 2x
y' = 1x3 - 2x⋅(3x2 - 2)
= 3x2 - 2x3 - 2x
Close
Example
Example
y = x3 ln x
y' = ?
Solution y' = ?
y = x3 ln x
y' = 3x2 ln x + x3⋅1x - [1]
= 3x2 ln x + x2
= x2(3 ln x + 1)
y' = 3x2 ln x + x3⋅1x - [1]
= 3x2 ln x + x2
= x2(3 ln x + 1)
Close
Example
y = xx
y' = ?
Solution y' = ?
y = xx
ln |y| = ln |xx|
ln |y| = x ln |x| - [1]
1y⋅y' = 1⋅ln |x| + x⋅1x - [2] [3]
y'y = ln |x| + 1
y' = y(ln |x| + 1)
= xx(ln |x| + 1)
ln |y| = ln |xx|
ln |y| = x ln |x| - [1]
1y⋅y' = 1⋅ln |x| + x⋅1x - [2] [3]
y'y = ln |x| + 1
y' = y(ln |x| + 1)
= xx(ln |x| + 1)
[1]
Close
Derivative of loga x
Formula
[loga x]' = 1x ln a
[loga |x|]' = 1x ln a
Example
y = log2 (x3 - 8x)
y' = ?
Solution y' = ?
y = log2 (x3 - 8x)
y' = 1x3 - 8x⋅(3x2 - 8) - [1]
= 3x2 - 8(x3 - 8x) ln 2
y' = 1x3 - 8x⋅(3x2 - 8) - [1]
= 3x2 - 8(x3 - 8x) ln 2
Close
Example
y = log5 5x6
y' = ?
Solution y' = ?
y = log5 5x6
= log5 5 + log5 x6
= 1 + 6 log5 |x| - [1]
y' = 0 + 6⋅1x ln 5
= 6x ln 5
= log5 5 + log5 x6
= 1 + 6 log5 |x| - [1]
y' = 0 + 6⋅1x ln 5
= 6x ln 5
[1]
Close
Derivative of ex
Formula
[ex]' = ex
Example
y = x2ex
y' = ?
Solution y' = ?
y = x2ex
y' = 2x1ex + x2ex - [1]
= 2xex + x2ex
= ex(x2 + 2x) - [2]
y' = 2x1ex + x2ex - [1]
= 2xex + x2ex
= ex(x2 + 2x) - [2]
Close
Example
y = ex2 + 4x
y' = ?
Solution y' = ?
y = ex2 + 4x
y' = ex2 + 4x⋅(2x1 + 4) - [1]
= ex2 + 4x(2x + 4)
= (2x + 4)ex2 + 4x
y' = ex2 + 4x⋅(2x1 + 4) - [1]
= ex2 + 4x(2x + 4)
= (2x + 4)ex2 + 4x
Close
Derivative of ax
Formula
[ax]' = ax ln a
Example
y = 22x + 1
y' = ?
Solution y' = ?
y = 22x + 1
y' = 22x + 1 ln 2⋅(2 + 0) - [1]
= 22x + 1⋅2 ln 2
= 22x + 2 ln 2 - [2]
y' = 22x + 1 ln 2⋅(2 + 0) - [1]
= 22x + 1⋅2 ln 2
= 22x + 2 ln 2 - [2]
Close
Derivative of sin x
Formula
[sin x]' = cos x
Example
y = x sin x
y' = ?
Solution y' = ?
Example
y = sin x2
y' = ?
Solution y' = ?
Example
y = sin2 x
y' = ?
Solution y' = ?
y = sin2 x
= (sin x)2
y' = 2(sin x)1⋅cos x
= 2 sin x cos x
= sin 2x - [1]
= (sin x)2
y' = 2(sin x)1⋅cos x
= 2 sin x cos x
= sin 2x - [1]
Close
Derivative of cos x
Formula
[cos x]' = -sin x
Example
y = cos (x3 - 4)
y' = ?
Solution y' = ?
y = cos (x3 - 4)
y' = -sin (x3 - 4)⋅(3x2 - 0) - [1]
= -3x2 sin (x3 - 4)
y' = -sin (x3 - 4)⋅(3x2 - 0) - [1]
= -3x2 sin (x3 - 4)
Close
Derivative of tan x
Formula
[tan x]' = sec2 x
Example
y = x2 tan x
y' = ?
Solution y' = ?
y = x2 tan x
y' = 2x1 tan x + x2 sec2 x - [1]
= 2x tan x + x2 sec2 x
y' = 2x1 tan x + x2 sec2 x - [1]
= 2x tan x + x2 sec2 x
Close
Derivative of csc x
Formula
[csc x]' = -csc x cot x
Example
y = csc (1 - x2)
y' = ?
Solution y' = ?
y = csc (1 - x2)
y' = -csc (1 - x2) cot (1 - x2)⋅(0 - 2x1) - [1]
= -2x csc (1 - x2) cot (1 - x2)
y' = -csc (1 - x2) cot (1 - x2)⋅(0 - 2x1) - [1]
= -2x csc (1 - x2) cot (1 - x2)
Close
Derivative of sec x
Formula
[sec x]' = sec x tan x
Example
y = sec3 x
y' = ?
Solution y' = ?
Derivative of cot x
Formula
[cot x]' = -csc2 x
Example
y = 7x cot x
y' = ?
Solution y' = ?
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. cosh x = ex - e-x2
(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 y' = ?
Derivative of cosh x
Formula
[cosh x]' = sinh x
Example
y = cosh2 x
y' = ?
Solution y' = ?
y = cosh2 x
= (cosh x)2
y' = 2(cosh x)1⋅sinh x - [1]
= 2 sinh x cosh x
= (cosh x)2
y' = 2(cosh x)1⋅sinh x - [1]
= 2 sinh x cosh x
Close