Product and quotient rule of derivative

The product of two functions can be differentiated using the product derivation rule

Example 1

Find the value of the derivative of the function h when x = 1

Instead of opening the parentheses, we denote

Now we can use product rule of derivative

The quotient of two functions can be differentiated using the quotient derivation rule

Example 2

Find the extremes value points of the function h

The denominator has no zeros, so the function is defined everywhere. We differentiate the function using the quotient derivation rule.

The zeros of the rational function are determined by the zeros of the numerator. The zero points of the numerator are x = -1 and x = 3. These are the extreme points of the function. The sign of the derivative is determined by the numerator because the denominator is positive for all values of the variable x.

The graph of the numerator is a downward-opening parabola. Let us make a behaviour chart of the function h.

The function has a local minimum at -1 and a local maximum at 3

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