Appendix: algebra and calculus basics









CONTINUITY AND DIFFERENTIABILITY

The derivative of logx. w.r.t. x is. 1 x. ; i.e.. 1. (log ) d x dx x. = . 5.1.12 Logarithmic differentiation is a powerful technique to differentiate 
leep


Dimensions of Logarithmic Quantities

to imply that log (x) is itself dimensionless whatever the dimensions of x. Differentiating eq 5 with respect to temperature gives.


New sharp bounds for the logarithmic function

5 mar. 2019 In this paper we present new sharp bounds for log(1 + x). We prove ... and


Compiled & Conducted @ JKSC

d (log x) = 1 dx x d (ex ) = ex dx d (ax) = ax.loga dx d (sin x) = cos x dx d (cos x) = ans : dy = x + 3 + (2x + 3). log x ... Differentiating wrt x ;.
MATH CH DIFFERENTIATION OTHER COLLEGE





Chapter 8 Logarithms and Exponentials: logx and e

x. = d dx log x. Then log xy and log x have the same derivative from which it follows by the Corollary to the Mean Value Theorem that these two functions 
chapter


Appendix: algebra and calculus basics

28 sept. 2005 6. The derivative of the logarithm d(log x)/dx
algnotes


DIFFERENTIAL EQUATIONS

An equation involving derivative (derivatives) of the dependent variable with Now substituting x = 1 in the above
leep


University of Plymouth

25 mai 2005 For the second part x2 is treated as a constant and the derivative of y3 with respect to y is 3y2. Exercise 1. Find. ∂z. ∂x and. ∂z. ∂ ...
PlymouthUniversity MathsandStats partial differentiation





Economics

Differentiation is all about measuring x x y y. −. − gives slope of the line connecting 2 points (x1 ... NOTE: the derivative of a natural log.
topic


RS Aggarwal Solutions Class 12 Maths Chapter 10 - Differentiation

RS Aggarwal Solutions for Class 12 Maths Chapter 10 -. Differentiation. 4. (i) tan (logx). (ii) log sec x. (iii) log sin x/2. Solution: 
RS Aggarwal Solutions Class Maths Chapter Differentiation


213588 Appendix: algebra and calculus basics

Appendix: algebra and calculus basics

c ?2005 Ben Bolker

September 28, 2005

1 Logarithms

Logarithms are the solutions to equations likey=exory= 10x.Natural logs, ln or log e, are logarithms basee(e= 2.718...);commonlogs, log10, are typically logarithms base 10. When you see just log it"s usually in a context where the difference doesn"t matter (although inRlog10islog10and logeis log).

1. log(1) = 0. Ifx >1 then log(x)>0, and vice versa. log(0) =-∞(more

or less); logarithms are undefined forx <0.

2. Logarithms convert products to sums: log(ab) = log(a) + log(b).

1

3. Logarithms convert powers to multiplication: log(an) =nlog(a).

4. You can"t do anything with log(a+b).

5. Converting bases: log

x(a) = logy(a)/logy(x). In particular, log10(a) = log e(a)/loge(10)≈loge(a)/2.3 and loge(a) = log10(a)/log10(e)≈log10(a)/0.434. This means that converting between log bases just means multiplying or dividing by a constant. You can prove this relationship as follows: y= log10(x) 10 y=x log e(10y) = loge(x) yloge(10) = loge(x) y= loge(x)/loge(10) (compare the first and last lines).

6. The derivative of the logarithm,d(logx)/dx, equals 1/x. This is always

positive forx >0 (which are the only values for which the logarithm means anything anyway).

7. The fact thatd(logx)/dx >0 means the function ismonotonic(always

either increasing or decreasing), which means that ifx > ythen log(x)> log(y) and ifx < ythen log(x)2 Differential calculus

1. Notation: differentation of a functionf(x) with respect toxcan be writ-

ten, depending on the context, as dfdx ;f?;f; orfx. I will stick to the first two notations, but you may encounter the others elsewhere.

2. Definition of the derivative:

dfdx = limΔx→0f(x+ Δx)-f(x)(x+ Δx)-x= limΔx→0f(x+ Δx)-f(x)Δx.(1) In words, the derivative is the slope of the line tangent to a curve at a point, or the "instantaneous" slope of a curve. The second derivative, d

2f/dx2, is the rate of change of the slope, or the curvature.

3. The derivative of a constant (which is a flat line if you think about it as

being a curve) is zero (zero slope).

4. The derivative of a line,y=ax, is the slope of the line,a.

5. Derivatives of polynomials:

d(xn)dx =nxn-1. 2

6. Derivatives of sums:

d(f+g)dx =dfdx +dgdx (andd(? iyi)/dx=? i(dyi/dx)).

7. Derivatives times constants:

d(cf)dx =cdfdx , ifcis a constant (dcdx = 0).

8. Derivative of the exponential:

d(exp(ax))dx =aexp(ax), ifais a constant. (If not, use the chain rule.)

9. Derivative of logarithms:

d(log(x))dx =1x

10. Chain rule:

d(f(g(x)))dx =dfdg

·dgdx

(thinking about this as "multiplying frac- tions" is a good mnemonic but don"t use that in general!)Example: d(exp(x2))dx =d(exp(x2))d(x2)·dx2dx = exp(x2)·2x.(2) Another example: people sometimes express the proportional change inx, (dx/dt)/x, asd(log(x))/dt. Can you see why?

11.Critical points(maxima, minima, and saddle points) of a curvefhave

df/dx= 0. The sign of the second derivative determines the type of a critical point (positive = minimum, negative = maximum, zero = saddle).

3 Partial differentiation

1. Partial differentiation acts just like regular differentiation except that you

hold all but one variable constant, and you use a curly d∂instead of a regular d. So, for example,∂(xy)/∂(x) =y. Geometrically, this is taking the slope of a surface in one particular direction. (Second partial derivatives are curvatures in a particular direction.)

2. You can do partial differentiation multiple times with respect to different

variables: order doesn"t matter, so ∂(f)∂(x)∂(y)=∂(f)∂(y)∂(x).

4 Integral calculus

For the material in this book, I"m not asking you to remember very much cal- culus, but it would be useful to remember that

1. the (definite) integral off(x) fromatob,?b

af(x)dx, represents the area under the curve betweenaandb; the integral is a limit of the sum?b x i=af(xi)Δxas Δx→0.

2. You can take a constant out of an integral (or put one in):

?af(x)dx= a?f(x)dx.

3. Integrals are additive:

?(f(x) +g(x))dx=?f(x)dx+?g(x)dx. 3

5 Factorials and the gamma function

Afactorial, written as (say)k!, meansk×k-1×...1. For example, 2! = 2,

3! = 6, and 6! = 720 (inRa factorial isfactorial()- you can"t use the

shorthand ! notation, especially since!=means"not equal to". Factorials come up in probability calculations all the time, e.g. as the number of permutations withkelements. Thegamma function, usually written as Γ (gamma()inR) is a generalization of factorials. For integers, Γ(x) = (x-1)!. Factorials are only defined for integers, but for positive, non-integerx(e.g. 2.7), Γ(x) is still defined and it is still true that Γ(x+ 1) =x·Γ(x). Factorials and gamma functions get very large, and you often have to com- pute ratios of factorials or gamma functions (as in the binomial coefficient, k!/(N!(N-k)!). Numerically, it is more efficient and accurate to compute the logarithms of the factorials first, add and subtract them, and then exponenti- ate the result: exp(logk!-logN!-log(N-k)!).Rprovides the log-factorial

Appendix: algebra and calculus basics

c ?2005 Ben Bolker

September 28, 2005

1 Logarithms

Logarithms are the solutions to equations likey=exory= 10x.Natural logs, ln or log e, are logarithms basee(e= 2.718...);commonlogs, log10, are typically logarithms base 10. When you see just log it"s usually in a context where the difference doesn"t matter (although inRlog10islog10and logeis log).

1. log(1) = 0. Ifx >1 then log(x)>0, and vice versa. log(0) =-∞(more

or less); logarithms are undefined forx <0.

2. Logarithms convert products to sums: log(ab) = log(a) + log(b).

1

3. Logarithms convert powers to multiplication: log(an) =nlog(a).

4. You can"t do anything with log(a+b).

5. Converting bases: log

x(a) = logy(a)/logy(x). In particular, log10(a) = log e(a)/loge(10)≈loge(a)/2.3 and loge(a) = log10(a)/log10(e)≈log10(a)/0.434. This means that converting between log bases just means multiplying or dividing by a constant. You can prove this relationship as follows: y= log10(x) 10 y=x log e(10y) = loge(x) yloge(10) = loge(x) y= loge(x)/loge(10) (compare the first and last lines).

6. The derivative of the logarithm,d(logx)/dx, equals 1/x. This is always

positive forx >0 (which are the only values for which the logarithm means anything anyway).

7. The fact thatd(logx)/dx >0 means the function ismonotonic(always

either increasing or decreasing), which means that ifx > ythen log(x)> log(y) and ifx < ythen log(x)2 Differential calculus

1. Notation: differentation of a functionf(x) with respect toxcan be writ-

ten, depending on the context, as dfdx ;f?;f; orfx. I will stick to the first two notations, but you may encounter the others elsewhere.

2. Definition of the derivative:

dfdx = limΔx→0f(x+ Δx)-f(x)(x+ Δx)-x= limΔx→0f(x+ Δx)-f(x)Δx.(1) In words, the derivative is the slope of the line tangent to a curve at a point, or the "instantaneous" slope of a curve. The second derivative, d

2f/dx2, is the rate of change of the slope, or the curvature.

3. The derivative of a constant (which is a flat line if you think about it as

being a curve) is zero (zero slope).

4. The derivative of a line,y=ax, is the slope of the line,a.

5. Derivatives of polynomials:

d(xn)dx =nxn-1. 2

6. Derivatives of sums:

d(f+g)dx =dfdx +dgdx (andd(? iyi)/dx=? i(dyi/dx)).

7. Derivatives times constants:

d(cf)dx =cdfdx , ifcis a constant (dcdx = 0).

8. Derivative of the exponential:

d(exp(ax))dx =aexp(ax), ifais a constant. (If not, use the chain rule.)

9. Derivative of logarithms:

d(log(x))dx =1x

10. Chain rule:

d(f(g(x)))dx =dfdg

·dgdx

(thinking about this as "multiplying frac- tions" is a good mnemonic but don"t use that in general!)Example: d(exp(x2))dx =d(exp(x2))d(x2)·dx2dx = exp(x2)·2x.(2) Another example: people sometimes express the proportional change inx, (dx/dt)/x, asd(log(x))/dt. Can you see why?

11.Critical points(maxima, minima, and saddle points) of a curvefhave

df/dx= 0. The sign of the second derivative determines the type of a critical point (positive = minimum, negative = maximum, zero = saddle).

3 Partial differentiation

1. Partial differentiation acts just like regular differentiation except that you

hold all but one variable constant, and you use a curly d∂instead of a regular d. So, for example,∂(xy)/∂(x) =y. Geometrically, this is taking the slope of a surface in one particular direction. (Second partial derivatives are curvatures in a particular direction.)

2. You can do partial differentiation multiple times with respect to different

variables: order doesn"t matter, so ∂(f)∂(x)∂(y)=∂(f)∂(y)∂(x).

4 Integral calculus

For the material in this book, I"m not asking you to remember very much cal- culus, but it would be useful to remember that

1. the (definite) integral off(x) fromatob,?b

af(x)dx, represents the area under the curve betweenaandb; the integral is a limit of the sum?b x i=af(xi)Δxas Δx→0.

2. You can take a constant out of an integral (or put one in):

?af(x)dx= a?f(x)dx.

3. Integrals are additive:

?(f(x) +g(x))dx=?f(x)dx+?g(x)dx. 3

5 Factorials and the gamma function

Afactorial, written as (say)k!, meansk×k-1×...1. For example, 2! = 2,

3! = 6, and 6! = 720 (inRa factorial isfactorial()- you can"t use the

shorthand ! notation, especially since!=means"not equal to". Factorials come up in probability calculations all the time, e.g. as the number of permutations withkelements. Thegamma function, usually written as Γ (gamma()inR) is a generalization of factorials. For integers, Γ(x) = (x-1)!. Factorials are only defined for integers, but for positive, non-integerx(e.g. 2.7), Γ(x) is still defined and it is still true that Γ(x+ 1) =x·Γ(x). Factorials and gamma functions get very large, and you often have to com- pute ratios of factorials or gamma functions (as in the binomial coefficient, k!/(N!(N-k)!). Numerically, it is more efficient and accurate to compute the logarithms of the factorials first, add and subtract them, and then exponenti- ate the result: exp(logk!-logN!-log(N-k)!).Rprovides the log-factorial
12345 Next
  1. log x differentiation formula
  2. log x differentiation by first principle
  3. log x differentiate
  4. log x derivative
  5. log x derivative by first principle
  6. log x derivative formula
  7. log x derivative proof
  8. log 1/x differentiation