Flashcards in 4.3.1 An Introduction to the Chain Rule Deck (15):
intro to chain rule
- A composite function is made up of layers of functions inside of functions. Some techniques of differentiation become very cumbersome when applied to composite
- The chain rule states that if f ( x ) = g ( h ( x )) , where g and h are differentiable functions, then f is differentiable and f ′ ( x ) = g ′ ( h ( x )) ⋅ h ′ ( x ) .
- A composite function is a function that results
from applying a function to the results of another
- Each different function that is applied can be
thought of as a layer of the composite function.
- To find the derivative of a composite function, you
must look at each layer.
- The chain rule is a shortcut for finding the
derivative of a composite function. The chain rule
must be used for each layer of the composite
- The chain rule states that the derivative of a
composition of two functions is equal to the
derivative of the outer function evaluated at the
inner function times the derivative of the inner
- Consider the inside of the composite function as a
“blop.” Take the derivative of that piece as though
the “blop” was just x. Then multiply that result by
the derivative of the “blop.”
- Notice that the chain rule can simplify the process of
finding some derivatives.
Suppose f(x)=(x^2−6)^4 / 8. Find f′(2).
Find the derivative of f(x).f(x)=(x^3+5x+1)^3
Find the derivative of:f(x)=3√x^4/3+x^1/3
Suppose f(x)=[x^2−(1+x^2)^2]^3. Find f′(x).
Find the derivative of:
h(x) = (3x + 5)^4
Suppose f(x)=(x^2−3)2(x^2+1)^2. Find f′(1).
Suppose f(x)=[2x−(1+x^2)^1/2]^2. Find f′(x).
Find the derivative of f(x).
Find the derivative.f(x)=(x^2+4)^2+(x^3+4x)^2
Find the derivative.f(x)=(x^2+1)^2
Find the derivative of P(t).P(t)=−(2t^2−14t+4)^2