Download y is explicit function of x if given y = f(x)

IMPLICIT DIFFERENTIATION
Terminology:
• y is explicit function of x if given y = f (x);
• y is implicit function of x if given g(x, y) = 0;
dy
• implicit differentiation finds dx
given g(x, y) = 0.
Steps for Implicit Differentiation:
dy
for finding dx
given equation with x and y.
• differentiate both sides of equation with respect to x,
assuming y is function of x;
dy
.
• solve equation for dx
Examples
1. find
2. find
dy
dx
for
dy
dx
for
6x2 + 8xy + y 2 = 6;
x4y 3 + 4x3/2 = 6y 3/2 + 5;
3. find tangent line at (16, 2) for curve determined by
√
2
2y − x = 4;
4. find tangent line at x = 2 for curve determined by
y 3 + xy 2 + 1 = x + 2y 2;
5. find
dR
dw
given
log(R(w)) = 1.83 − 0.43 log(w).
RELATED RATES
Related Rates Problems:
given an equation with two (or more) variables, find the
relation between the rates of change for the variables.
Steps for Solving Related Rates Problems:
• identify variables and draw sketch, if possible;
• write equations relating the variables ;
• find derivatives relations using implicit differentiation;
• solve for unknown derivative, subsitute given values.
Examples
• assuming functions x(t) and y(t), find
dy
dt
if
dx
xy−5x+2y = 2, with
= −5, when x = 2, y = −3;
dt
2
• assuming functions x(t) and y(t), find
dy
dt
if
dx
= 6, when x = 1, y = 0;
dt
• spherical snowball, radius r = 4in, melting with radial
change of .25in/hour: find rate of change of volume;
• sliding 25ft ladder has base moving at 4ft/min: find
rate of change of top when base is 7ft from wall;
• brain weight b of fetus with weight w: if b = 0.22w0.87,
db
find dw
dt , if dt = .25g/day, when w = 25g;
• crime rate C with temperature T when T = 76oF : if
dT
o
C = (T − 60)2/10 + 100, find dC
given
dt
dt = 8 F/m.
xey = 1 + ln(x), with
2