Ch 3. Derivatives II ma Multimedia Engineering Math Higher OrderDerivatives RelatedRates Differential Newton’sMethod
 Chapter 1. Limits 2. Derivatives I 3. Derivatives II 4. Mean Value 5. Curve Sketching 6. Integrals 7. Inverse Functions 8. Integration Tech. 9. Integrate App. 10. Parametric Eqs. 11. Polar Coord. 12. Series Appendix Basic Math Units Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Hengzhong Wen Chean Chin Ngo Meirong Huang Kurt Gramoll ©Kurt Gramoll

MATHEMATICS - THEORY

In this section the higher order derivatives of a function and the method to calculate them are discussed.

Higher Order Derivatives

 Name Notation First Derivative , , , , Second Derivative , , , , Third Derivative , , ,, nth Derivative , , , ,

When more than one derivative is applied to a function, it is consider higher order derivatives. For example, the derivative of y = x7 is 7x6. 7x6 is differentiable and its derivative is 42x5. 42x5 is called the second order derivative of y with respect to x for function x7. Repeating this process, function y's third order derivative 210x4 is obtained. The higher order derivative notations is shown on the left.

The chain rule also holds for higher order derivatives. For instance, one can find the fourth derivative for
y = A sin(ωt + 1), in which y is a function of t.

Consider y is a function of θ where θ = ωt +1. Therefore, the first order derivative is

The second order derivative is

The third order derivative is

The forth order derivative is

The implicit differentiation method also holds for higher order derivatives. For example, one can find the second derivative for ellipse .

In ellipse, y is a function of x. In order to find the derivative of y with respect to x, implicit differentiation method is applied. The first derivative is

(1)

Rearranging equation (1) gives

(2)

Taking another derivative with respect to x gives

(3)

Substitute equation (2) to (3),

(4)