Still More Examples of the Chain Rule

Assume 

• f(x) =  ln(x) implies f '(x) = 1 /x 
• g(x)= exp(x)  = e^x   implies g '(x) = exp(x) =  e^x . 

Examples in the following Real player webvideos which illustrate the chain rule with trigonometric, logarithmic and exponential functions. Bon Appetit.

1. [Play Video] 2½  minutes: (i) Derivatives of ln(x), e^x, cos(x) and sin(x) and (ii) Chain Rule for general outer functions.
   
2. [Play Video] 2½ minutes: Chain Rule Examples with y = sin(3x) and y = ln( x²+1). 
   
3. [Play Video]  3½ minutes: More Chain Rule Examples - cases where the chain rule is applied separately to terms in a  sum.
   
4. [Play Video]  2¼ minutes: Derivatives of ln(x), a^x  and 5^x using the formulas  a^x  = e^{x ln(a)}  and the chain rule when a >0  is not a function of x.  Exercise find the derivative of g(x)^f(x) =  e^f(x)ln(g(x)) using the chain rule twice. Assume g(x) > 0 for all x.

Calculus Appetizers
& Lessons

Starter Guide (Views)
Real Player Videos

3. Derivative Motivation
3.  Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x)  (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3.  Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots

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