Understanding Calculus

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  Home
  Testimonials
  Table of Contents

  Preface
  1. Why Study
  Calculus
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  Motion
  8. Understanding
  Derivative
  9. Derivative
  Approximations
  10. Integration
  Theory
  11. Understanding
  Integration
  12. Differentials

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

  Links
  Contact

 
 
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CHAPTER 5

Chapter 5 - Differentiating Functions

Section 5.5 - Differentiating Functions of Functions

The last technique of differentiation is for differentiating functions of functions of x or functions of the form, . For example consider the function:

The derivative is not because we also need to take into consideration the inside function of x, . We can replace with g(x) and get:

To find the derivative of with respect to x, we first need to find the derivative of f with respect to g. From the definition of the derivative:

Next we find the derivative of g(x), the inside function, with respect to x.

Now we multiply the two derivatives to get df / dx :

goes to zero, or:

Similarly, also goes to zero as goes to zero. Multiplying both sides by dx.

Thus as dx goes to zero;

As dx goes to zero, becomes . At the same time the in the denominator of cancels out with the in the numerator of , since they are both the equivalent. To conclude

Thus the derivative of a function of a function of x with respect to x, is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function with respect to x. Some examples will show how this is done.

Next section -> Section 6.1 - Motion, Velocity and Acceleration

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