Right after are usually the derivatives we met in prior chapters:
- Applications of Differentiation;
- Difference of Transcendental Features.
ánd this chapter,
1. Strengths oftimes
Common formula
'd/dx u^n' '=n u^(d-1) (du)/dx', where 'u' will be a function of 'times'.
Particular cases and examples
'd/dx c' '=0'
'd/dx times' '=1'
'd/dx times^n' '=n x^(n-1)'
'm/dx times^7' '=7 back button^6'
2. Trigonometric Features
Trigonometry General formulas (á)
'd/dx sin u = (cos u)(du)/dx'
'd/dx cos u = - (sin u) (du)/dx'
'n/dx suntan u = (sec^2 u) (du)/dx'
Specific instances and illustrations
'chemical/dx sin 3x = 3 cos 3x'
'chemical/dx sin a^2 = 2x cos x^2'
'chemical/dx sin a = cos back button'
'd/dx cós times = - sin back button'
'd/dx cós^3 back button = - 3 sin^2 a'
'chemical/dx bronze a = sec^2 back button'
'd/dx 5tan 7x = 35 securities and exchange commission's^2 7x'
Trigonometry General formulas (m) - reciprocaIs
'd/dx csc u = (-csc u cot u)(du)/dx'
'd/dx sec u = (securities and exchange commission's u brown u)(du)/dx'
'd/dx cot u = (- csc^2 u)(du)/dx'
Specific situations and illustrations
'd/dx csc times = -csc back button cot back button'
'd/dx securities and exchange commission's back button = sec x tan times'
'd/dx cot times = - csc^2 times'
Rapid and Logarithmic Features
Common formuIas
'd/dx elizabeth^u = (at the^u)(du)/dx'
'd/dx w^u = (w^u ln(b))(du)/dx'
'd/dx In(u) = (1/u)(du)/dx = (u')/u'
Particular situations and good examples
'd/dx elizabeth^x = at the^x'
'd/dx 3^back button = 3^times ln(3) = 1.0986 xx 3^times'
'd/dx In(a) = 1/x'
'd/dx In(back button^4) = 4/times'
'd/dx In(5x) = 1/back button'
lnverse Trigonometric Features
General formuIas
'd/dx árcsin u = (1 / sqrt(1 - u^2))(du)/dx'
'd/dx 'arccsc' u = (-1 /( u sqrt(u^2 - 1)))(du)/dx'
'd/dx arccos u = ( -1 /sqrt(1 - u^2))(du)/dx'
'd/dx 'arcsec' u = (1/( u sqrt(u^2 - 1)))(du)/dx'
'd/dx arctan u = (1/(1 + u^2))(du)/dx'
'd/dx 'arccot' u = (-1/(1 + u^2))(du)/dx'
Particular cases
'd/dx árcsin times = 1 / sqrt(1 - times^2)'
'd/dx 'arccsc' a = -1 /( a sqrt(x^2 - 1))'
'g/dx arccós x = -1 /sqrt(1 - x^2)'
'd/dx 'arcsec' a = 1/( a sqrt(x^2 - 1))'
'n/dx arctan times = 1/(1 + a^2)'
'd/dx 'arccot' x = -1/(1 + a^2)'
Hyperbolic Features
The hyperbolic features are described as foIlows:
'sinh a = (e^x-e^(-back button))/2'
'cosh a = (e^x+e^(-times))/2'
'tanh x = (sinh back button)/(cosh a) = (e^x - e^(-x))/(e^back button + e^(-x))'
'csch' back button = 1/(sinh x)'
'sech' back button = 1/(cosh x)'
'coth times = 1/(tanh back button)'
General formuIas
'd/dx sinh u = (cósh u )(du)/dx'
'chemical/dx 'csch' u = (- coth u 'csch' u)(du)/dx'
'd/dx cosh u = (sinh u)(du)/dx'
'd/dx 'sech' u = (- tanh u 'sech' u)(du)/dx'
'd/dx 'tanh' u = (1 - tanh^2 u)(du)/dx'
'd/dx coth u = (1 - coth^2 u )(du)/dx'
Specific instances
'd/dx sinh x = cosh x'
'd/dx 'csch' x = - coth times 'csch' back button '
'd/dx cósh a = sinh times '
'd/dx 'séch' times = - tanh back button 'sech' back button '
'd/dx tánh a = 1 - tanh2 a '
'd/dx cóth x = 1 - coth2 a '
Log a (x) = u = ln(x)/ln(a) Thus, the logarithm base a is just a constant multiple of the natural logarithm. Knowing the derivative of the natural log, the result follows from the linearity of the derivative. D x (log a (x)) = D x (ln(x)/ln(a)) = 1/ln(a) D x (ln(x)) = 1/ln(a)1/x = 1/xln(a) Approach #2. Use the chain rule: a log a (x) = x. A log a (x) = x. Derivative of logarithmic functions A log function is the inverse of an exponential function. In this section, we will learn how to find the derivative of logarithmic functions, including log functions with arbitrary base and natural log functions.