Flashcards in 4.2.1 The Product Rule Deck (12):
• The derivative of a product of two functions is not necessarily equal to the product of the two derivatives.
• The product rule states that if p(x) = f(x)g(x), where f and g are differentiable functions, then p is differentiable and
p'(x) = f(x)g'(x) + g(x)f'(x)
- To find the derivative of a product of two functions you can first simplify the functions and then find the derivative of the result.
- At first glance, there does not seem to be any pattern to the derivative.
- But if you decompose the product just right, you can see that the derivative is actually made up of four pieces: the original two functions and their derivatives.
- The product rule is a shortcut for taking the derivative of a product of two functions. Remember the product rule by using the chant to the left.
- The derivative of a product of two functions is equal to the first times the derivative of the second plus the second times the derivative of the first.
- Expanding this function would take a lot of work. But the product rule makes finding the derivative easy.
- You might need to do some side calculations to use the
- Remember the chant! It is a simple way to remember the product rule.
Find the derivative. f(x)=(x+1)(x^2+1)
Find the derivative.f(x)=(x+1)(x−2)
Find the derivative of F(x) given that F(x)=f(x)⋅g(x),f(x)= 3x^2+1, and g(x)=√x.
Suppose f (x) = (2x ^2 − 1) (x + 1). Which of the following lines is tangent to f and parallel to the line y = x + 2?
y = x + 1
Suppose f (x) = (x^ 2 − 2x) (2x − x ^2 ). At which point is the tangent line horizontal?
Find the derivative.h(x)=(2x^4+3x+7)(x^5−3x^2)
Suppose f (x) = (x^ 2 − 2x) (3x + 2).
What is the equation of the line normal to f
(i.e., the line perpendicular to the tangent line)
at the point (1, −5)?
Suppose f (x) = (3x − x ^2 ) (2x − x ^2 ). Find the equation of the line tangent to f at the point (1, 2).
y = x + 1
Suppose f (x) = (4x^ 3 + 3) (1 − x^ 2 ). What is the equation of the line tangent to f at the point (1, 0)?
y = −14x + 14