The Most Important Derivatives and Antiderivatives:

The Sum Rule, the Constant Multiple Rule and the Power Rule for Integration:Sum Rule for Integration – integrate long expressions term by term. Expressed formally:

Constant Multiple Rule for Integration – move a constant outside of an integral before you integrate. Expressed in symbols:

Power Rule for Integration – integrate any real power of x (except -1). Expressed formally:

where n ≠ –1,

Solve Integrals with Variable Substitution:1. Declare a variable u, set it equal to an algebraic

expression that appears in the integral, and then substitute u for this expression in the integral.

2. Differentiate u to find dudx

, then isolate all x

variables on one side of the equal sign.3. Make another substitution to change dx and all

other occurrences of x in the integral to an expression that includes du.

4. Integrate by using u as your new variable of integration.

5. Express this answer in terms of x.

Integration by Parts:Gives you the option to break down product of two functions to its factors and integrate in altered form. Follow these steps:1. Decompose the entire integral (including dx) into

two factors.2. Let the factor without dx equal u and the factor

with dx equal dv.3. Differentiate u to find du; integrate dv to find v.4. Use the formula: 5. Evaluate the right side of this equation to solve the

integral.

Compound Functions Where Inner Function is ax + b :Integrate the outer function f, and the inner function g(x) is of the form ax + b — it differentiates to a constant. Here are some examples:

Compound Functions Where Inner Function is ax:Integrate the outer function f, and the inner function g(x) is of the form ax — it differentiates to a constant. Here are some examples:

Chapter 7: Transcendental FunctionsSection 7.1 – Inverse Functions and Their Derivatives: One-to-One functions – f(x) is one-to-one on a

domain D if f(x1) ≠ f(x2) whenever x1 ≠ x2 in D.Function y = f(x) one-to-one if graph intersects each horizontal line at most once.

Inverse functions – suppose f is one-to-one function on domain D with range R, inverse function f -1 defined by f -1(b) = a if f(a) = b. Domain of f -1 is R and range is D.

Finding inverse – passing from f to f -1 summarized as two-step procedure:1. Solve equation y = f(x) for x, giving x = f -1(y) where x is function of y.2. Interchange x and y, giving formula y = f -1(x).

Derivatives of inverses of differentiable functions – derivatives are reciprocals of one another slopes

are reciprocal: (f -1)’(b) = 1f'(a)

or (f -1)’(b) =

1f'( f-1(b)¿

¿.

Derivative rule for inverses – if f has interval I as domain and f’(x) exists and is never zero on I, then f -1 is differentiable at every point in its domain (range of f). Value (f -1)’ at point b in domain of f -1 is reciprocal of value of f’ at point a = f -1(b):

(f -1)’(b) = 1f'( f-1(b)¿

¿ or

d f-1

dx∨ ❑

x = b=1dfdx

∨ ❑x = f-1(b)

.

Section 7.2 – Natural Logarithms: Definition of natural logarithm function – natural

logarithm of any positive number x, written as ln x, is defined as integral. Natural logarithm is function

given by ln x = ∫-1

x 1t

dt , x > 0.

Number e is number in the domain of natural

logarithm satisfying ln(e) = 1 ∫1

e 1t

dt = 1.

Derivative of y = ln x: ddx

ln x =

ddx ∫

1

x 1t

dt = 1x

for every positive value x, ddx

ln x = 1x

,

chain rule extends formula for positive functions u(x):

ddx

ln u = ddu

ln u ∙dudx

so…

ddx

ln u = 1u

dudx

, u > 0

ddx

ln|1x|, x ≠ 0

Properties of logarithms1. Product rule: ln bx = ln b + ln x

2. Quotient rule: ln bx

= ln b – ln x

3. Reciprocal rule: ln 1x

= -ln x

4. Power rule: ln x r = r ln x

The integral ∫1u

du = ln |u| + C if u = f(x),

then ∫ f'(x)f(x)

dx = ln |f(x)| + C whenever

f(x) differentiable function that is never zero.

∫sin x dx = -cos x + C

∫ tan x dx = -ln |cos x| + C

∫cos x dx = sin x + C ∫cot x dx =

ln |sin x| + C

∫sec x dx = ln |sec x + tan x| + C

∫csc x dx = -ln |csc x + cot x| + C

Section 7.3 – Exponential Functions: Inverse of ln x and the number e – for every real

number x, we define natural exponential function to be e x = exp x.Inverse equations for e x and ln x:e ln x = x (all x > 0) and… ln (e x) = x (all x)

Derivative and integral of e x : if u is any

differentiable function of x, then ddx

e u = e u

dudx .

General antiderivative of the exponential function:

∫e udu = e u + C

For all numbers x, x1 and x2, the natural exponential e x obeys the following laws:

1. e x1∙e x2 = e x1+x 2 3. e x1

e x2 =

e x 1−x2

2. e-x = 1e x 4. (ex1) r = e r x1, if r is

rational The general exponential function ax – for any

numbers a > 0 and x, the exponential function with base a is: a x = e x ln a when a = e, the definition

gives a x = e x ln a = e x ln e = ex ∙ 1 = e x in particular,

a n ∙ a -1 = a n-1

Proof of the power rule (general version) – for any x > 0 and for any real number n, x n = e n ln x General power rule for derivatives – for x > 0 and

any real number n, ddx

x n = nx n-1 ; if x ≤ 0, then

the formula holds whenever the derivative, x n, and

x n-1 all exist so… ddx

x n = ddx

e n ln x = nx n-1

Differentiate f(x) = x x, x > 0:

f’(x) = ddx

(e x ln x) = e x ln x ddx

(x ln x) = e x ln x (ln x +

x ∙1x

)

= x x (ln x + 1) The derivative of a u :

ddx

a x = ddx

e x ln a = ddx

(x ln a) = a x ln a

if a = e, then ln a = 1 so…

derivative of a x simplifies to ddx

e x = e x ln e = e x

If a > 0 and u is differentiable function of x, then a u

is differentiable function of x and ddx

a u = a u ln a

dudx

∫ a u du =

a u

ln a + C

Logarithms with base a – for any positive number a ≠ 1, log a x is the inverse function of a x

When a = e, log e x = inverse of e x = ln x.

Rules for base a logarithms – for any numbers x > 0 and y > 01. Product rule: log a xy = log a x + log a y

2. Quotient rule: log a

xy

= log a x – log a y

3. Reciprocal rule: log a 1y

= -log a y

4. Power rule: log a x y = y log a x Inverse equations for a x and log

a x:

a loga x = x (x > 0) and… log a x (ax) = x (all x)

log a x = ln xln a

ddx

log a u = ddx

(ln uln a

) = 1ln a

ddx

(ln u)

Derivatives and integrals involving log a x:

ddx

(log a u) = ddx

(ln uln a

) = 1ln a

ddx

(ln u) = 1ln a

∙ 1u

dudx

ddx

(log a u) = 1ln a

∙ 1u

dudx