Sentence examples for derivative of equation from inspiring English sources

To solve for α, we calculate the derivative of Equation 12 w.r.t α and equate the result to zero as follows: ∂E ∂α = - 2 ( G r - I r - α ( F r - I r ) ) ( F r - I r ) - 2 ( G g - I g - α ( F g - I g ) ) ( F g - I g ) - 2 ( G b - I b - α ( F b - I b ) ) ( F b - I b ) (19).

The conditions for the objective function to be convex if β ≠ 1 (case B), can be derived as follows: First, take the derivative of equation (14) with respect to n d : This expression will always be positive for β ≥ 2, and hence the objective function (equation (13)) convex.

We compute the derivative of equation (2.6) with respect to x to derive the following higher order partial differential equation: frac{partial^{k}}{partial x^{k}} bigl{ G t,x a,b,q) bigr} = ( tlog a ) ^{k}G t,x a,b,q), where k is a nonnegative integer.

The partial derivative of Equation 3 with respect to t was set to zero to derive Equations 4 and 5.

The first derivative of equation (1) expresses smoothness, and the subtracted part expresses observation error.

In fact, by taking the temporal derivative of Equation 1, we obtain: Q = A h dh dt (2).

Taking the partial derivatives of Equation A1 with respect to, α k, gives (A8) (A8).

It is straightforward to show that the signs of the partial derivatives of equation (11) are the same as those of equation (9).

A more formal analysis of partial derivatives of equation (7) with respect to different growth parameters is given in Appendix A.1.

We take the homogeneous balance between nonlinear terms and highest order derivatives of equation (6) to determine the positive integer n.

After making the tedious partial derivatives of equation (34), the solution of the unknown parameters (a 1,a 2,…,a k ) and (b 1,b 2,…,b k ) can be derived.