Basic Chain Rule
The derivative of 4(cos x)2
-8sinx
Average ROC for f(x) = 2x2 - 4 over the interval [2, 5]
(show set up)
Aver ROC = [f(5) - f(2)]/(5 - 2) = (46 - 4)/3 = 14
The derivative of x(sin x)
(1)(sin x) + (x)(cos x)
The derivative of (cos x)/(sin x)
[(sin x)(sin x) - (cos x)(-cos x)]/(sin x)2
The derivative of 3{(x + 1)2 - 4/(x3 - 2x)}
3{2(x + 1)(1) + [4/(x3 - 2x)2][3x2 - 2]}
The derivative of (2x2 + 3x - 2)2
(2x2 + 3x - 2)(4x + 3)
Average ROC for f(x) = 3x3+4 over the interval [-1, 2]
(show set up)
Aver ROC = [f(2)-f(-1)]/(2+1) = [28-1]/3 = 9
The derivative of x²cosx
(2x)(cos x)+ (x²)(-sin x)
The derivative of 2/(x+1)
-2/(x+1)²
The derivative of tan(sinx)
[sec²(sin x)][cos x]
The derivative of ln(x2 - x)
[1/(x2 - x)][2x - 1]
Instantaneous ROC for f(x) = (3x2 + 1)2 @ x = 2
f'(x) = 2(3x2 + 1)(6x)
f'(4) = 2[(3)(4) + 1][(6)(2)] = 312
The derivative of y = x³lnx
y′ = (3x²)(ln x)+(x3)(1/x)
The derivative of (1+lnx)/(x²-lnx)
[(1/x)(x2-lnx) - (1+lnx)(2x-1/x)] / (x²-lnx)²
The derivative of ln[(sin x)/x]
[x/(sin x)]{[(cos x)(x) - (sin x)(1)]/x2}
The derivative of 5(tan x)3
15(tan x)2(sec x)2
Instantaneous ROC for f(x) = 1/(x3 - 2x) @ x = 3
f(x) = (x3 - 2x)-1
f'(x) = [(-1)(x3 - 2x)-2][3x2 - 2]
f'(3) = (-25)/(21)2 = -25/441
The derivative of x3/2(ex)
(3/2 x1/2)(ex) + (x3/2)(ex)
The derivative of ex/(tan x)
[(ex)(tan x) - (ex)(sec2x)]/(tan2x)
The derivative of [10x][e(2x - lnx)]
[10][e(2x - lnx)] + [10x][e(2x - lnx) (2 - 1/x)]
The derivative of e(x^2 - 3x + x^3)
e(x^2 - 3x + x^3)(2x-3+3x2)
Instantaneous ROC for f(x) = exlnx @ x =1
f'(1) = (e1)(ln(1)) + (e1)(1/(1)) = e
The derivative of (x2 - 3)(5x - x3 + sin x)
(2x)(5x - x3 + sin x) + (x2 - 3)(5 - 3x2 + cos x)
The derivative of (x³lnx)/(x+2)
{[(3x2)(ln x) + (x3)(1/x)](x+2)-(x3lnx)(1)}/(x+2)²
The derivative of ln[tan2x]
[tan2x]-1[2(tan x)(sec2x)]