Another way of computing this derivative is to view the composite function as the composite of and h. Applying the chain rule to this situation gives:

Applying the above function to the arguments ( 5, 2 ), we have:

* Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is

* Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven:

Applying the axiom of choice again we get a function from the first order formulas to such functions

Applying this function to our current state we obtain

Applying a result of MacIntyre on the model theory of p-adic integers, one deduces again that ζ < sub > G </ sub >( s ) is a rational function in p < sup >− s </ sup >.

Applying the Klein-Gordon operator on both sides, remembering that the four-moment is on-shell and that is the Green's function of the operator, we obtain:

Applying Bayesian probability in practice involves assessing a prior probability which is then applied to a likelihood function and updated through the use of Bayes ' theorem.

Applying these substitutions, the quadratic function ( 2 ) is transformed into

Applying these substitutions, we obtain the parametric function:

Applying the state-transition function to a state may yield two possible results: either an element of together with the next state of the stream, or the element of the singleton set 1 as a separate " final state " indicating that there are no more values in the stream.

Applying Kummer's 24 = 6 × 4 transformations to the hypergeometric function gives the 6

Applying this rule of thumb, one can apply " common sense " when creating the duration of a single activity or group of activities necessary to produce a deliverable defined by the WBS.

Applying the Curry – Howard isomorphism once more, inductive families correspond to inductively defined relations.

Applying the above bundle map ( pointwise ) to this section yields the pullback of α by φ, which is the 1-form φ < sup >*</ sup > α on M defined by

Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities.

As defined above one can also get degree of freedom using minimum number of coordinates required to specify a position. Applying it:

Applying an annihilation operator followed by its corresponding creation operator returns the number of particles in the k < sup > th </ sup > single-particle eigenstate:

Applying a negative DC voltage (" bias ") to the control grid will repel some of the electron stream back towards the cathode, thus isolating the plate from the cathode ; full bias will turn the tube off by blocking all current from the cathode.

Applying vinegar to common jellyfish stings deactivates the nematocysts ; however, placing the affected areas in hot water is a more effective treatment because the venom is deactivated by heat.

Applying methods first suggested by Finlay, yellow fever was eradicated in Cuba and later in Panama, allowing completion of the Panama Canal.

Applying a conception of such illocutionary acts according to which they are ( roughly ) acts of saying something with the intention of communicating with an audience, he describes indirect speech acts as follows: " In indirect speech acts the speaker communicates to the hearer more than he actually says by way of relying on their mutually shared background information, both linguistic and nonlinguistic, together with the general powers of rationality and inference on the part of the hearer.

Applying ROT13 to a piece of text merely requires examining its alphabetic characters and replacing each one by the letter 13 places further along in the alphabet, wrapping back to the beginning if necessary.

Applying sound changes apparent in other Sogdian words and inherent in Indo-European he traces the development of * Suγδa from Skuda, " archer ," as follows: Skuda > * Sukuda by anaptyxis > * Sukuδa > * Sukδa ( syncope ) > * Suγδa ( assimilation ).

Applying thick wraps to both of his knees became a pre-game ritual, and by the end of his career, simply swinging a bat caused him to fall to one knee in pain.

Applying the cosmological principle, this suggests that heavier elements were not created in the Big Bang but were produced by nucleosynthesis in giant stars and expelled across a series of supernovae explosions and new star formation from the supernovae remnants, which means heavier elements would accumulate over time.

Applying the method to hydrogen, they found that the moment of a proton was 2. 785 ± 0. 02 nuclear magnetons, and not 1 as predicted by the current theory, while that of a deuteron was 0. 855 ± 0. 006 nuclear magnetons.

Applying for a commercial pilot's licence on 2 June 1921 ( in which he gave his name as ' Charles Edward Kingsford-Smith '), he became one of Australia's first airline pilots when he was chosen by Norman Brearley to fly for the newly formed West Australian Airways.

Applying the results of local studies to the global scale are complicated by the role the ocean's circulation plays in different ocean regions.

** Applying styles by outline level ( e. g., bold all 1st level items )

Applying CGT to a position attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position.

Applying procedural fairness to an entire society therefore seems certain to create recognizable inefficiencies, and therefore be unfair, and ( by the equivalence of justice with fairness ) unjust.

* Free Advice On Applying To Graduate School ( PDF )-Written by a professor of psychology at Portland State University.

) Applying this to the geometric progression 31, 62, 124, 248, 496 ( which results from 1, 2, 4, 8, 16 by multiplying all terms by 31 ), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31, 62, 124, 248.

Applying this principle to the art of poetry, and analysing, line by line and even word by word, the works of great poets, he deduced the law that the beauty of poetry consists in the accuracy, beauty and harmony of individual expression.

Applying these methods in the workplace, such as Csikszentmihalyi did with Swedish police officers, can improve morale by fostering a sense of greater happiness and accomplishment, and in correlated to increased performance.

Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v ; the use of the scalar Laplacian and vector Laplacian gives two more:

Applying Itō's lemma with f ( S ) =

Applying Itō's lemma with f ( Y ) = log ( Y ) gives

Applying both sides to e < sub > j </ sub >, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).

Applying the Lagrange interpolation formula to do quadratic interpolation on the inverse of f yields