Derivatives, Derivatives & More Derivatives

Basic

100

y=x^2+4sqrtx

y'=2x+2x^(-1/2) 

200

y=(2-x)/(3x+4)

y'=− 10/( 3 x + 4 )^2

300

y= (x − 1 )^4 ( x + 1 )^3

y'=( x − 1 )^3 ( x + 1 )^ 2 ( 7 x + 1 )

400

y=xln(4x-1)^3

y'=12x/(4x-1)+ln(4x-1)^3

500

y=ln(e^(4)/(5x-1))

y'=-5/(5x-1)

Trig

100

y=tan(2x)

y'=2sec^2(2x)

200

y=csc^5(1-4x)

y'=20csc^5(1-4x)cot(1-4x)

300

y=arctan(x/2)

y'=2/(4+x^2)

400

y=arcsin(e^(6x))

y'=(6e^(6x))/sqrt(1-e^(12x))

500

y=cot(log(5^(2x-1)))

y'=-csc^2(log(5^(2x-1)))(2ln5)/(ln10)

Implicit

100

y^2-x^2=4

dy/dx=x/y

200

y+cot^(-1)(y)+x-4=0

y'=(-y^2-1}/y^2

300

4x^2-6xy^3+y=10

y'=(-8x+6y^3)/(1-18xy^2)

400

Find y''

y+siny=5x

(d^2y)/(dx^2)=(25siny)/(1+cosy)^3

500

Find y''

e^ln(5)-8^{1-y}-7x=9

(d^2y)/(dx^2)=(49(8^(2y-2)))/(3ln2)

Logarithmic Diff

100

y=x^(4x)

y'=x^(4x)(4+4lnx)

200

y=(secx)^x

y'=(secx)^x[ln(secx)+xtanx]

300

y=x^sqrtx

y'=x^sqrt(x)(lnx/(2sqrt(x))+1/sqrt(x))

400

y=((x+1)^4sqrt(x^2-5))/7^(2x-1)

y'=(4/(x+1)+x/(x^2-5)-2ln7)y

500

y=sqrt((x+1)sqrt((x+2)sqrt((x+3)sqrt((x+4)

y'=[1/(2(x+1))+1/(4(x+2))+1/(8(x+3))+1/(16(x+1))]y

Extra

100

Find

(f^(-1))'(0) ... when ... f(x)=cscx-sqrt(2

-1/sqrt2

200

Find: 

(f^(-1))'(-3) ... when ... f(x)=x^2+4x

1/2,-1/2

300

Find the instantaneous rate of change of the function at x = 1.

f(x)=sin^2((pix)/4)

f'(1)=pi/4

400

Find the x-value(s) where the tangent to the graph is parallel to the x-axis.

f(x)=3x^4-5x^3+2

x=0, 5/4

500

Find the equation of the tangent at (0, 1) of the curve,

y^3-xy^2+cos(xy)=2

y=1+1/3x