Example 1: Derivative of a Function to the Fourth Power Find the derivative of the function (d/dx) 3x4 using the Constant Multiple Rule. Solution Apply the Constant Multiple Rule by taking the derivative of the power function first and then multiply with the coefficient 3. (d/dx) 3x4 =...
Applying the Constant Multiple Rule Find the derivative of g(x)=3x2g(x)=3x2 and compare it to the derivative of f(x)=x2f(x)=x2. Show Solution Applying Basic Derivative Rules Find the derivative of f(x)=2x5+7f(x)=2x5+7. Show Solution Find the derivative of f(x)=2x3−6x...
The rule that the derivative of a constantonlyapplies if you take the derivative of a constant (aka apolynomial functionof zeroth degree), and not constants that also have exponents, constants multiplied by x, or anything other than a number. While √9 is a constant, √9x is not. If in...
where η is viscosity, the subscript (n) denotes the nth order derivative on time, and n = 0.5 in Eq. (51). This equation can be considered as the initial fractional constitutive model for the viscoelastic solid material. Considering the advantages of the fractional models after the work don...
Because the velocity of the fluid particles varies in a linear manner with respect to the y coordinate, it is clear that V/b = dvx/dy, which is the derivative of the velocity with respect to the distance y. Equation (1) can be rewritten as: (2)τyx=−μ(dυx/dy). This states...
Fill in the blanks: a. If c is a constant , then \frac {d}{dx}(c) = \underline{ \ \ \ \ \ \ \ \ }, b. The Power Rile states that is n is any real number, then \frac {d}{dx}(x^n) = \underline{ \ \ \ \ \ \ \ \ },...
stress tensorwith respect to the mean effective stress, yielding the stress ratio tensor. This accounts for the property of granular materials that their strength and stiffness strongly depend on the pressure level. The rate of effective stress is then decomposed by the chain rule of differentiation...
The point {eq}A \left ( 6,3 \right ) {/eq} maps onto {eq}{A}' \left ( 2,1 \right ) {/eq} under a dilation with respect to the origin. What is the constant of dilation? a. {eq}\frac{1}{3} {/eq} b. {eq}\frac{1}{2} ...
Now, at any position xi (or simply i), the second derivative in Equation 20-10 can be approximated using Equation 20-4, that is, (20-14)d2p(xi)/dx2 = {pi-1 − 2pi+ pi+1}/(Δx)2+O(Δx)2=0 so that the finite difference model for our differential equation ...
In recent decades, much research has been done in managed pressure drilling, especially in CBHP, including proportional–integral–derivative (PID) control, linear model predictive control (MPC), nonlinear MPC, adaptive control, and linear-quadratic-gaussian control. Nonlinearity manifests in the ...