Note2: (arcsin u ) ' = negative of (arccos u ) '
#DERIVATIVE OF LOG BASE 3 PLUS#
Note1: Arc tan 's derivative is the only one with no root and with a plus sign Just as in the trig function derivatives, du represents the derivative of the argument function. Level 3: something ( 7 x) gives us (something)' (7 - in front!)Ģ0(3) cos 3 (3 x+1) sin (3 x+1) (chain rule)Ħ tan 3 x sec 2 3 x sin 5 x + 5cos 5 x tan 2 3 x (product & chain rule)Ģ[≥csc 4 x (csc x cot x)sec 2 3 x + 6 sec 3 x (sec 3 x tan3 x) csc 5 x Level 2: sin ( something) gives us cos ( something) Level 1: thing cubed gives us 3( thing ) 2 We're using the chain rule on 3 levels of functions: The derivative of y = sin 3 ( 7 x) is y ' = ( 3 sin 2 ( 7 x ))( 7 cos 7 x)ģ sin 2 ( 7 x ) is the derivative of ( sin 7 x ) 3ħ cos 7 x is du (7) times the derivative of sin 7 x (cos 7 x ).
If y = sin 3 (7 x) we're really indicating that y = (sin 7 x) 3 however, this notation causes confusion since it looks like the (7 x) is cubed also - when it's not! Note: the notation for powers of the trig functions can be confusing.
Then ( co sec u ) ' = du co sec u co tan u Just add the minus sign and the "co" syllable. Note 2: Notice the symmetry of the pairs. Note 1: If the function's name includes the "co" syllable (ex: cotan), the derivative is negative! Otherwise, we may be tempted to multiply the du by the argument - and we mustn't do that. And, since we can't change the argument of the trig function, we should write du at the beginning of the expression for the derivative. If the argument of the trig function is u = f ( x ), then the derivative must include du, the derivative of the argument.
#DERIVATIVE OF LOG BASE 3 HOW TO#
Most texts teach us how to differentiate trig functions when the argument (angle) is just x. So, if h ( x ) = ( something ) n, h '( x ) = n ( something ) n 1 (something ) ' We're dealing strictly with the function f ( x ) - once that's done - we find g '( x ) and multiply. Notice that when we find f '( x ) - on the outside layer - we leave g ( x ) alone. Inside thing = g( x) = ( 3 x 2 + 5 x 1 ). It deals with layers of functions - so pretend to peel the layers like peeling an onion.ġst: take f ', the outside function's derivative - leave inside alone!Ģnd: multiply by derivative of inside function. In the example above: u = x 3 + 5 and v = 2 x + 1ĭon't be fooled into thinking that is a quotient! The numerator is a constant! Not a function, therefore rewrite the function as 7( x + 3) 1, then use the chain rule.Ī fraction like, so it too is not a quotient.Ĭhain rule: This is the one that causes lots of headaches. HINT: do the " v 2 " part first or you'll forget it! The numerator function is u and the denominator function is v.
Quotient rule:so named since it's used on a quotient of 2 or more functions. In the example above: u = 6 x 2 and v = x 8 One of the functions is u and the other is v. Product rule: so named since it's used on a product of 2 or more functions. Power rule: applies to any term of the form ax n where n is a constant no matter if n is positive, negative, rational, or not, as long as it includes no variables. When we need to find a higher derivative (2nd, 3rd, etc.) the notation is similar to that for the first derivative - but eventually, the "primes" become too numerous - so we use either brackets around a number or Roman numerals to indicate the level of differentiation. That is, or, to represent the change in volume per unit time. In Related Rates problems, we should always use the format that signifies slope: Most calculus "doers" end up using y ' to denote the first derivative - but, as mentioned above, there are other symbols which at times are more suited to the question. Here are the different ways of denoting the first derivative. Since the derivative represents the slope of the tangent, the best notation is because it reminds us that the derivative is a slope =. There are many ways to denote the derivative, often depending on how the expression to be differentiated is presented.