Chapter 3 Differentiation Rules (continued)

Chapter 3 Differentiation Rules (continued)

Sec 3.5: Implicit Differentiation (continued)

Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph of a function? x 2 + y 2 = 25 at (-3, 4)

Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph of a function? x 3 + y 3 = 6xy at (3, 3) Can x 3 + y 3 = 6xy be solved for y in the vicinity of (3, 3)?

Implicit Differentiation When an equation in x and y is solved for y, i.e. is written as y = stuff with x s only (e.g. y = x 3 + sinx ), then y is defined explicitly as a function of x. When an equation in x and y is not solved for y, (e.g. ysin(x) = x or sin xy = x + e y2 ), then y is defined implicitly as a function of x.

Note: = d dx Implicit Differentiation

Implicit Differentiation Problems Ex 3: For the curve defined by x 3 + y 3 = 6xy, a) Find dy dx b) Find the equation of the tangent line to the curve at (3, 3) c) Find the points on the curve that have a horizontal tangent line d) Find the points on the curve that have a vertical tangent line e) What s going on at the point (0, 0)? f) Find d2 y dx 2

Implicit Differentiation Problems Ex 4: For the curve defined by e x/y + y 2 tan(y 2 ) = sin xy + 7x y + 1, find dy dx.

Derivatives of Elementary Functions

Derivatives of Exponential Functions (Sec. 3.4)

Derivatives of Exponential Functions Ex 5: Derive the formula for d dx 2x

Derivatives of Exponential Functions Rule for the derivative of an exponential function: d dx ax = a x lna

Derivatives of Exponential Functions Quickies Ex 6: Find a) d dx 7x = 7 x ln (7) b) 10 x = 10 x ln (10) c) d dt 4 4 4 3 t = 3 t ln 3

Derivatives of Inverse Trig. Functions (Sec. 3.5)

Derivatives of Inverse Trig Functions Ex 7: Derive the formula for d dx sin 1 x

Derivatives of Inverse Trig Functions Ex 8: Derive the formula for d dx tan 1 x

Derivatives of Inverse Trig Functions Formulas d dx sin 1 x = 1 d 1 x 2 dx cos 1 x = 1 1 x 2 d dx tan 1 x = 1 d 1 + x 2 dx cot 1 x = 1 1 + x 2 d dx sec 1 x = 1 x x 2 1 d dx csc 1 x = 1 x x 2 1

Derivatives of Log Functions (Sec. 3.6)

Derivatives of Log Functions Ex 9: Derive the formula for d dx log 2 x

Derivatives of Log Functions Rule(s) for the derivative of an log function: In particular d dx log a x = 1 xlna d dx ln x = 1 x and d dx ln x = 1 x

Derivatives of Log Functions Quickies Ex 10: Find a) d dx log 12 x = 1 xln12 b) log x = 1 xln10 c) d dx ln x = 1 x d) ln x = 1 x

Derivatives of Hyperbolic Functions (Sec. 3.11)

Some Things to Know About Hyperbolic Functions Recall the definitions of the hyperbolic functions: sinh x = ex e x 2 tanh x = coth x = cosh x = ex + e x sinh x cosh x = ex e x e x + e x cosh x sinh x = ex + e x e x e x 2

Some Things to Know About Hyperbolic Functions Recall the definitions of the hyperbolic functions: sinh x = ex e x 2 cosh x = ex + e x 2 sech x = 1 cosh x = 2 e x e x csch x = 1 sinh x = 2 e x + e x

Some Things to Know About Hyperbolic Functions Graphs: f x = sinh x Domain =, Range =, Resitricted Domain for sinh 1 x =,

Some Things to Know About Hyperbolic Functions Graphs: f x = cosh x Domain =, Range = 1, ) Resitricted Domain for cosh 1 x = 0, )

Some Things to Know About Hyperbolic Functions Graphs: f x = tanh x Domain =, Range = 1, 1 Resitricted Domain for tanh 1 x =,

Some Things to Know About Hyperbolic Functions Identities: sinh x = sinh(x) cosh x = cosh (x) cosh 2 x sinh 2 x = 1 1 tanh 2 x = sech 2 x sinh x + y = sinh x cosh y + cosh x sinh y cosh x + y = cosh x cosh y + sinh x sinh y

Derivatives of Hyperbolic Functions Ex 11: Derive the formula for d sinh x dx

Derivatives of Hyperbolic Functions Ex 12: Derive the formula for d tanh x dx

Derivatives of Hyperbolic Functions Formulas d dx sinh x = cosh x d dx cosh x = sinh x d dx tanh x = sech2 x d dx coth x = csch2 x d dx sech x = sech x tanh x d csch x = csch x coth x dx

Derivatives of Inverse Hyperbolic Functions (Sec. 3.11)

Some Things to Know About Inverse Hyperbolic Functions Recall the formulas for the inverse hyperbolic functions: sinh 1 x = ln x + x 2 Domain = + 1, cosh 1 x = ln x + x 2 1 Domain = 1, ) tanh 1 x = 1 2 ln 1 + x 1 x Domain = 1, 1

Some Things to Know About Inverse Hyperbolic Functions Graphs: f x = sinh 1 x Domain =, Range =,

Some Things to Know About Inverse Hyperbolic Functions Graphs: f x = cosh 1 x Domain = 1, ) Range = 0, )

Some Things to Know About Inverse Hyperbolic Functions Graphs: f x = tanh 1 x Domain = 1, 1 Range =,

Derivatives of Inverse Hyperbolic Functions Ex 13: Derive the formula for d dx cosh 1 x (in 2 ways)

Derivatives of Inverse Hyperbolic Functions Formulas d dx sinh 1 x = 1 d 1 + x 2 dx cosh 1 x = 1 x 2 1 d dx tanh 1 x = 1 d 1 x 2 dx coth 1 x = 1 1 x 2 d dx sech 1 x = 1 x 1 x 2 d dx csch 1 x = x 1 x 2 + 1

Mixed Examples (Sec. 3.1-3.6, 3.11)

Examples Ex 14: Find y if a) y = ln cosh x + 4 x tan 1 (x) b) y = arctan 1 x 1+x c) y = log 3 (cos x) sinh 1 x 5 xlnx

Derivatives of Inverse Trig Functions Ex 15: Find dy dx if tan 1 x 2 y = x + xy 2