## Understanding Calculus

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 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

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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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