Differentiating Functions Using the Chain Rule
Differentiating Functions Using the Chain Rule
Understanding the Chain Rule
- Know that the chain rule is a fundamental tool used in calculus for differentiating functions of functions.
- Clarify that the chain rule states that the derivative of a composed function is the derivative of the outer function times the derivative of the inner function.
- Memorise that the chain rule formula in its basic form can be written as
(f(g(x)))' = f'(g(x)) * g'(x).
Applying the Chain Rule
- Learn to identify when a function requires the chain rule for differentiation, typically when one function is embedded within another.
- Practice applying the chain rule to a variety of functions like exponential, logarithmic, and trigonometric.
- Recognise that the chain rule can be applied multiple times in scenarios where a function is made up of more than two functions.
The Chain Rule with Trigonometric Functions
- Become comfortable with applying the chain rule to trigonometric functions like sine, cosine and tangent.
- Know the basic derivatives of sin(x), cos(x) and tan(x), which are cos(x), -sin(x) and sec^2(x) respectively.
- Understand how the chain rule can be used in conjunction with the rest of your trigonometric knowledge to differentiate more complex trigonometric functions.
Solved Problems and Examples
- Work through a number of example problems to solidify your understanding on the application of the chain rule.
- Tackle simple to complex tasks that fully test your proficiency and ability in differentiating functions using the chain rule.
- Use solutions to sample problems as a guide to understanding the process of applying the chain rule effectively and correctly.
Practical Applications of the Chain Rule
- Explore practical real-world problems that can be solved by applying the chain rule.
- Recognise the importance of the chain rule in physical and geometrical problems that involve the relationship between different variables.
- Understand how the chain rule can be used in conjunction with other differentiation methods and tools in solving sophisticated mathematical problems.