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Chapter Contents

• Differentiation
• 1. Limits and Differentiation
• 2. The Slope of a Tangent to a Curve (Numerical)
• 3. The Derivative from First Principles
• 4. Derivative as an Instantaneous Rate of Change
• 5. Derivatives of Polynomials
• 6. Derivatives of Products and Quotients
• 7. Differentiating Powers of a Function
• Differentiation Java applet
• Derivative of a Cubic - Exploration using GeoGebra
• 8. Differentiation of Implicit Functions
• 9. Higher Derivatives
• 10. Partial Derivatives

• Comments, Questions?

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From the math blog...

9. Higher Derivatives

by M. Bourne

We can continue to find the derivatives of a derivative. We find the

• second derivative by taking the derivative of the first derivative,
• third derivative by taking the derivative of the second derivative... etc

 

Example

If , then what are the higher derivatives?

Answer

Here are the LiveMath answers for this example.

LIVEMath

Application - Acceleration

We saw before that acceleration is the rate of change of velocity:

But we also know that velocity is the rate of change of displacement:

So it follows that the second derivative of displacement will give us acceleration:

Example

If the displacement (in metres) at time t (in seconds) of an object is given by

s = 4t³ + 7t² - 2t,

find the acceleration at time t = 10.

Solution:

s = 4t³ + 7t² - 2t

At t = 10, the acceleration will be

a = 24(10) + 14 = 254 ms^-2.

 

Higher Derivatives of Implicit Functions

Example

a. Find the second derivative of the implicit function xy + y² = 4.

Answer

b. Find the value of the second derivative of the implicit function in part (a) when x = 2, where y > 0.

Answer