Unit 3: Differentiation

Composite

100

Differentiate y = sin(3x)

dy/dx = 3cos(3x)

200

Differentiate y = ln(5x)

dy/dx = 1/x

300

Differentiate y = e^{tan x}

dy/dx = e^{tan x} * sec²x

400

Differentiate y = cos(4x)

dy/dx = -4sin(4x)

500

Differentiate y = ln(tan x)

dy/dx = 1/sin x cos x

Implicit

100

Differentiate 5x² + 2y³ = 4

dy/dx = -5x/3y²

200

Differentiate In(y3) = 5x + 3

dy/dx = 5y/3

300

Differentiate sin(xy) = x

dy/dx = 1-ycos(xy)/xcos(xy)

400

Differentiate sin(x + y) = 2x

dy/dx = 2sec(x+y)-1

500

Differentiate e^xy = x + y

dy/dx = 1-ye^xy / xe^xy - 1

inverse

100

If f(x) = ln x, find (f^-1})’(1)

Since f^-1(x) = e^x, (f^-1)’(1) = e¹ = e

200

If f(2) = 3 and f’(2) = 4, find (f^-1)’(3)

 (f^-1)’(3) = 1/f'(2) = 1/4

300

 If f(x) = e^x, find (f^-1)’(1)

Since f^-1(x) = ln x, (f^-1)’(1) = 1/1 = 1

400

If f(x) = tan x, find (f^-1)’(1)

Since f^-1(x) = arctan x, (f^-1)’(1) = 1/1+x² x = 1 = 1/2

500

Given g(x) = cos(x) + 3x², g(pi/2) = 3pi²/4, Find 

(g^-1)'(0)

1/3pi-1

Higher order derivatives

100

Given that y = sin x + ln(5x), find y''

y'' = -sinx - 1/x²

200

Given that y = e^xx, find y''

y'' = e^xx +2e^x

300

Given that y = sin²x, find y''

y'' = 2cos²x - 2sin²x

400

Given that dy/dx = y² + 2x - 1, find d²y/dx²

d²y/dx² = 2y³ + 4xy - 2y + 2

500

If f(x) = -3x³ + 4x^-2, find f' '(2)

75/2

Formulas

100

What is the Product rule?

d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

200

What is the power rule for differentiation?

d/dx (xⁿ) = nx^n-1

300

What is quotient rule?

d/dx[f(x)/g(x)] = f'(x)g(x)-f(x)g'(x)/[g(x)²]

400

What is chain rule?

d/dx[f(g(x))] = f'(g(x)) * g'(x)

500

What is the derivative of an inverse function f^-1(x)?

d/dx[f^-1(x)] = 1/f'(f^-1(x))