Chain Rule Examples

Understanding the chain rule is essential in differential calculus. It helps differentiate complex functions by breaking them down into simpler parts, like trigonometric, exponential, and logarithmic functions combined with polynomials. This guide covers various examples to clarify the process.

1. Trigonometric functions with inner polynomial functions

   • Apply the chain rule by differentiating the outer trigonometric function first.
   • Multiply by the derivative of the inner polynomial function.
   • Common examples include sin(ax + b) and cos(ax + b).

2. Exponential functions with inner polynomial functions

   • Differentiate the outer exponential function using the base e or other bases.
   • Multiply by the derivative of the inner polynomial function.
   • Examples include e^(ax + b) and a^(ax + b).

3. Logarithmic functions with inner polynomial functions

   • Use the chain rule to differentiate the outer logarithmic function.
   • Multiply by the derivative of the inner polynomial function.
   • Common forms include ln(ax + b) and log_a(ax + b).

4. Composite polynomial functions

   • Differentiate the outer polynomial function first.
   • Multiply by the derivative of the inner polynomial function.
   • Examples include (ax^2 + bx + c)^n.

5. Square root functions with inner polynomial functions

   • Differentiate the outer square root function using the power rule.
   • Multiply by the derivative of the inner polynomial function.
   • Common forms include √(ax^2 + bx + c).

6. Inverse trigonometric functions with inner polynomial functions

   • Differentiate the outer inverse trigonometric function.
   • Multiply by the derivative of the inner polynomial function.
   • Examples include arcsin(ax + b) and arccos(ax + b).

7. Rational functions with inner polynomial functions

   • Use the quotient rule in conjunction with the chain rule.
   • Differentiate the numerator and denominator separately.
   • Common forms include (ax + b)/(cx + d).

8. Power functions with inner polynomial functions

   • Differentiate the outer power function using the power rule.
   • Multiply by the derivative of the inner polynomial function.
   • Examples include (ax + b)^n.

9. Nested exponential functions

   • Differentiate the outer exponential function first.
   • Multiply by the derivative of the inner exponential function.
   • Common forms include e^(e^(ax + b)).

10. Combinations of trigonometric and exponential functions

    • Apply the chain rule to both the trigonometric and exponential components.
    • Differentiate each function in the composition and multiply by the derivatives of the inner functions.
    • Examples include e^(sin(ax + b)) and sin(e^(ax + b)).