Week 1 Flashcards
What is the product rule?
d/d(x) [f(x)g(x)] = fg’ + gf’
What is the quotient rule?
d/d(x) [f(x)/g(x)] = (gf’ - fg’)/g^2 or [lowd(high) - highd(low)] / (low sq’d)
What is the derivative of e^x?
e^x
What is the chain rule?
F(x) = f(g(x)) and F'(x) = f'(g(x)) * g'(x) y = f(u) and u = g(x) --> dy/dx = (dy/du) * (du/dx)
Rewrite b^x
(e^lnb)^x –> e^((lnb)*x)
What is the derivative of b^x
lnb * b^x
What is the derivative of sin?
cos
Rewrite cos^2(x)
(cos(x))^2
What’s the derivative of cos?
-sin
What’s the derivative of 2^x?
ln2 * 2^x
What’s the derivative of tanx?
sec^2 * x
How do you tell if f(x) and g(x) are inverses of each other?
If f(g(x)) = x
If f(x) and g(x) are inverses of each other, what is g’(x)?
1/[f’(g(x))]
What is the inverse of y = e^x
y = lnx
What is the derivative of lnx?
1/x
Rewrite y = logbasea x
lnx / lna
What is the derivative of lnx / lna?
1/(xlna)
When is implicit differentiation used?
When y is not explicitly defined, derivative of x and y can be used
What is the derivative of a constant equal to?
0
What is absolute extrema?
Max and min outputs on the entire interval
What is local extrema
All the max and min outputs along the interval
What is the slope of the tangent to line at the extrema?
0 or derivative does not exist
What is the critical number of a function?
x’ = 0 or DNE
Do all critical numbers imply a local min or max?
No
When is a curve decreasing or increasing?
Decreasing when y’ is neg; increasing when y’ is pos
What is concave up vs concave down?
Concave up is the curve as a valley with both sides continuing up (tangent lines below curve; f’’ is pos); concave down is the curve at a mountain top with both sides continuing down (tangent lines above curve; f’’ is neg)
What is an inflection point?
Point where the concavity changes (f’’ = 0)
What is the first derivative test?
See attached screenshot
Summarize the 2nd derivative test
f’‘>0, local min; f’‘<0 local max
