Quotient and Product Rule
Take the derivative.
y = (x-1)/(x+1)
y' = -2/(x+1)^2
What is the derivative of log base a of (x)?
1/[x*ln(a)]
Take the derivative.
y = (x^3-5x)^4
y' = 4(x^3 - 5x)^3 * (3x^2 - 5)
What is the power rule?
d/dx(x^n) = ?
d/dx(x^n) = nx^(n-1)
How do you make 7 even?
Take away the s.
Take the derivative.
y = (x^2 - 4x)*(5x^3 + 1)
y' = (15x^3)(x^2 - 4x) + (2x - 4)(5x^3 + 1)
What is the derivative of a^x?
a^x * ln(a)
Take the derivative.
y = e^(6x^3 - 7x)
y' = (e^(6x^3 - 7x))* (18x^2 - 7)
What is the derivative of e^x and lnx?
e^x and 1/x
What does the zero say to the infinity?
Nice belt.
Take the derivative.
y = (14x - 1)/(x^2+8)
y' = [(x^2 + 8)(14) - (14x-1)(2x)]/[(x^2+8)^2]
$10,000 is invested into an account with interest compounded continuously at an interest rate of 2.35%.
a. Write an equation modeling the amount in the account after t years.
b. Write an equation modeling the instantaneous rate of change of the amount in the account after t years.
a. A(t) = 10000*e^(0.0235t)
b. A'(t) = (0.0235)*10000*e^(0.0235t)
Take the derivative.
y = ln(5x^4 - 6x)
y' = (20x^3-6)/(5x^4 - 6x)
What is the product rule?
d/dx[f(x)*g(x)] = f(x)*g'(x) + g(x)*f'(x)
How do you make one burn?
Differentiate a log fire.
d/dx[log(fire)] = 1/(fire) *d/dx(fire)
Take the derivative.
y = (e^x)(lnx)
y' = (e^x)/x + (e^x)(lnx)
Find dy/dx if x^2 + y^3 = 2y
dy/dx = 2x/(2-3y^2)
Take the derivative.
y = 3^(9x^2 + 7x)
y' = 3^(9x^2 + 7x) * ln(3) * (18x + 7)
What is the quotient rule?
d/dx[f(x)/g(x)] = [g(x)*f'(x) - f(x)*g'(x)]/(g(x)^2)
What's the first derivative of a cow?
Prime rib.
Find the derivative.
y = e^x/(x^3-5x)
y' = (x^3 - 5x)(e^x) - (e^x)(3x^2-5)
(x^3 - 5x)^2
Air is being pumped leaking out of a spherical balloon at a rate of 6 cm^3/second. At what rate is the radius decreasing when the radius is 3 cm?
The radius is decreasing at a rate of 1/(6*pi) cm/sec. Or the rate of change of radius is - 1/(6*pi) cm/sec.
*Did you have negatives in your work?* Quantities were decreasing - you should have!
Take the derivative.
y = [(7x^4 - 3x)/(4x^2 + 5x)]^5
y'=5[(7x^4 - 3x)/(4x^2 + 5x)]^4 * [(4x^2 + 5x)*(28x - 3) - (7x^4 - 3x)*(8x + 5)]/[(4x^2 + 5x)]^2
What is the chain rule?
[f(g(x))] = f'(g(x))*g'(x)
What math is discussed between sea birds?
Inter-gull calculus.