Download 3_05_3_06_Implicit

Math 1304 Calculus I
3.5 and 3.6 – Implicit and Inverse Functions
Implicit and Explicit Functions
• Explicit: y = f(x)
• Implicit: F(x,y)=0
Example:
x  y 1
2
2
y  1 x
2
implicit
explicit
Implicit Differentiation
• If f(x) = g(x), then f’(x) = g’(x)
• Example: x2 + y2 = 1
Inverse
• f and g are inverse if:
y = f(x) iff x = g(y)
• Also f and g are inverse if
f(g(y) = y and g(f(x) = x
• Examples
Exponential and Log
y = ln(x) iff x = ey
Trigonometric: sin and arcsin
y = arcsin(x) iff x = sin(y)
Derivatives of inverse functions
d
1
arcsin( x) 
2
dx
1 x
d
1
arccos( x) 
dx
1 x2
d
1
arctan( x) 
2
dx
1 x
Proof? (in class)
Derivative of Logarithms
• If F(x) = loga(f(x)), then F’(x) = (1/ln a)
f’(x)/f(x)
• Proof? (in class)
• Special case:
If F(x) = ln(f(x)), then F’(x) = f’(x)/f(x)
A new good working set of rules
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Constants: If F(x) = c, then F’(x) = 0
Powers:
If F(x) = f(x)n, then F’(x) = n f(x)n-1 f’(x), where n is real
Exponentials: If F(x) = af(x), then F’(x) = (ln a) af(x) f’(x)
Logarithms: If F(x) = loga(f(x)), then F’(x) = (1/ln a) f’(x)/f(x)
Trigonometric functions:
If F(x) = sin(f(x)), then F’(x) = cos(f(x)) f’(x)
If F(x) = csc(f(x)), then F’(x) = - csc(f(x)) cot(f(x)) f’(x)
If F(x) = cos(f(x)), then F’(x) = - sin(f(x)) f’(x) If F(x) = sec(f(x)), then F’(x) = sec(f(x)) tan(f(x)) f’(x)
If F(x) = tan(f(x)), then F’(x) = sec2(f(x)) f’(x) If F(x) = cot(f(x)), then F’(x) = - csc2(f(x)) f’(x)
Inverse trig functions:
If f(x) = arcsin(x), then f’(x) = 1/√ (1-x2)
If f(x) = arccos(x), then f’(x) = -1/√ (1-x2)
If f(x) = arctan(x), then f’(x) = 1/(1+x2)
Scalar multiplication: If F(x) = c f(x), then F’(x) = c f’(x)
Sum:
If
F(x) = g(x) + h(x), then F’(x) = g’(x) + h’(x)
Difference: If
F(x) = g(x) - h(x), then F’(x) = g’(x) - h’(x)
Multiple sums: derivative of sum is sum of derivatives
Linear combinations: derivative of linear combination is linear combination of derivatives
Product:
If
F(x) = g(x) h(x),
then F’(x) = g’(x) h(x) + g(x)h’(x)
Multiple products: If
F(x) = g(x) h(x) k(x),
then F’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x)
Quotient:
If
F(x) = g(x)/h(x),
then F’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2
Composition: If F = fog is a composite, defined by F(x) = f(g(x))
then F'(x) = f'(g(x))g'(x)
Logarithmic Differentiation
• Sometimes it helps to take the ln of both
sides of an equation before differentiation.
• Then solve for y’
• Examples:
y = f(x)g(x)
Use of logarithmic differentiation
• Prove general power law
• Quick proof of product rule