Basic Rules of Derivatives Examples with Communicators Calculus.

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Presentation transcript:

Basic Rules of Derivatives Examples with Communicators Calculus

More Examples: Find You’ll notice none of the basic rules specifically mention radicals, so you should convert the radical to its exponential form, X 1/2 and then use the power rule.

More Examples: Find Again, you need to rewrite the expression so that you can use one of the basic rules for differentiation. If we rewrite the fraction as x -2,then we can use the power rule.

More Examples: Find Rewrite the expression so that you can use the basic rules of differentiation. Now differentiate using the basic rules.

Example 1: Use The Product Rule Find the derivative of.

Example 2 Differentiate the function:

Example 4 Differentiate the function: Use the QUIOTIENT RULE!

Example 3 If h(x) = x[g(x)] and it is known that g(3) = 5 and g'(3) =2, find h'(3). Find the derivative: Evaluate the derivative:

Example 5 Find an equation of the tangent line to the curve at the point (1,½). Find the Derivative Find the derivative (slope of the tangent line) when x=1 Use the point-slope formula to find an equation

Example 6 Differentiate the function: Sum and Difference Rules Constant Multiple Rule Power Rule Simplify The Quotient Rule is long, don’t forget to rewrite if possible. Rewrite to use the power rule Try to find the Least Common Denominator

Example 2 of the Product Rule: Find f’(x) for

Example of the Quotient Rule: Find First we will find the derivative by using The Quotient Rule

Example 2: Find the derivative of For this quotient doing the division first would require polynomial long division and is not going to eliminate the need to use the Quotient Rule. So you will want to just use the Quotient Rule.

Another example: Find all the x values where has a horizontal tangent line. Find the derivative. Since horizontal lines have a slope of 0, set the derivative equal to 0 and solve for x. Thus the x values where the function has horizontal tangents is at x = -1, -1/3.

Find the slope and equation of the tangent line to the curve at the point (1,3). First find the derivative of the function. Now, evaluate the derivative at x = 1 to find the slope at (1,3).

Example continued: Now we have the slope of the tangent line at the point (1,3) and we can write the equation of the line. Recall to write the equation of a line, start with the point slope form and use the slope, 4 and the point (1,3).