Derivative Rules PAL

Power Rule

100

If f(x) = 5x⁴ - 8, then f'(x) = __

What is 20x³?

200

Which country has the most time zones?

France (12 of them)

300

If f(x) = 3√x+⁵√x, then f'(x) = __

What is 3/2x^-1/2+1/5(x^-4/5)?

400

If f(x) = (4x³+3x²)/x, then f'(x) = __

What is 4x²+3x

500

If f(x) = x²+(3/4)², then f'(x) = __

What is 2x?

Quotient Rule

100

If f(x) = 5 / 4x³ + 4, then f'(x) = __

(Don't simplify denominator)

What is − 15x² / (4x³ + 4)²?

200

If y = 3x⁴ + 2 / 3x³ − 2, then y' = __

What is  9x⁶ − 24x³ − 18x² / (3x³ − 2)²?

300

If f(x) = e^2x / 2x⁴ − 4, then f'(x) = __

(Simplify denominator)

What is  e^2x(x⁴-2x³-2) / (2x⁴ − 4)²?

400

If y = e^x/(x+2), then y' = __

(Simplify denominator)

What is  e^x(x+1)/(x+2)²

500

If f (x) = 2 / x⁵ − 5, then f'(x) = __ 

(Don't simplify denominator)

What is − 10x⁴ / (x⁵ - 5)²?

Chain Rule

100

If f(x) = (5x² + 3)⁴, then f'(x) = __

What is 40x(5x² + 3)?

200

If f(x) = ln(e^sin(x)), then f'(x) = __

What is cos(x)

300

If f(x) = ((x + 5)⁵ − 1)⁴, then f'(x) = __

What is 20((x + 5)⁵ − 1) 3 ⋅ (x + 5)⁴?

400

If y = tan^-1(e^2x), then y' = __

(1/(1+e^4x))*2e^2x

500

If y = tan(ln(x³)), then y' = __

What is sec²(ln(x³))*3x/x³?

Product Rule

100

If f(x) = (2x⁴ − 3)(x² + 1), then f'(x) = __

What is 12x⁵ + 8x³ − 6x?

200

If f(x) = (5x⁵ + 5)(−2x⁵ − 3), then f'(x) = __

What is −100x⁹ − 125x⁴?

300

If y = (x⁴ + 3)(3x²+sin(x)), then f'(x) = __ 

f'(0)=

f'(x)=(4x³)(3x²+sin(x))+(x⁴ + 3)(6x+cos(x))

f'(0)= 0+(3)*(1)=3

400

If f(x) = e^xsin(x)+cos(x), then f'(0) = __

What is e^xsin(x)+e^xcos(x)-sin(x)

f'(0)= 1

500

If f(x) = √x*e^2x, then f'(x) = __

What is e^2x(2x^1/2+(1/2)*x^-1/2)

Miscellaneous

100

​​​​​​​​d/dx(sin(e^2x))

cos(e^2x)*2e^2x

200

If f(x)+x²[f(x)]³=10 and f(1)=2,

Find f'(1).

-16/13

300

d/dx (sin^-1( ln(2x)))

1/(x*√1-ln²(2x))

400

What gas do plants release during photosynthesis?

O₂

500

Find dy/dx if x⁹-xy⁷+y⁹=1

dy/dx= (y⁷-9x⁸)/(9y⁸-7xy⁶)