2.4 The Product and Quotient Rules
We saw in the last section that the derivative of a sum or difference is the sum or difference of the derivatives. However, we can NOT say that the derivative of a product (or quotient) is the product (or quotient) of its derivatives, as seen in the following examples:
Let f(x) = x2 x3
Let g(x) = x5/x2
The correct formula for calculating the derivative of a product was discovered by Leibniz and is called the Product Rule.
Or, in prime notation, (fg)' = f g' + g f ' = gf '+ fg'
Example: Find y' if y = (5x ­ 4)(8x + 3). 1
Example: Find f '(x) if f(x) = x3 sin x. Ex: Find g'(x)if g(x) = x ­3 √x
Find the slope of the tangent line and the slope of the normal line to the graph of g(x) at the point (4, 1/32).
Suppose we know that h(x) = f(x) g(x) and we know that f(4) = ­2, f '(4) = 6, g(4) = 5,
and g'(4) = ­3. Find h'(4). 2
Example: Find y' if y = (5x3 + 4x2 ­ 3x)(sin x + 8x­5)
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The following rule allows us to find the derivative of a quotient of two differentiable functions. Or, in prime notation, Ex:
Ex: Find y' for the following functions.
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5
Suppose we know that h(x) = and we know that g(4) = 5 and g'(4) = ­3. Find h'(4). Find an equation of the tangent line to the curve at the point (1, 2). Find the derivative of the function
At what points does the graph of f(x) have a horizontal tangent? 6
Trigonometric Functions Knowing the derivatives of sine and cosine allows us to find the derivatives of the other 4 trig functions by using the quotient rule.
Given y = tan x, find y'.
We could similarly find the derivatives of the other trig functions, which are listed below.
Find y' for each of the given functions.
y = 5cos x sin x
y = x2 csc x
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