Differentiationsregeln Pdf

Differentiating Complex Functions: The Chain and Product Rules

Understanding the Fundamentals

Differentiating complex functions involves applying the chain rule and the product rule, which are essential techniques for finding the derivatives of intricate functions.

Chain Rule

The chain rule is used when you have a function within a function. To find the derivative, you differentiate the outer function with respect to the intermediate variable and multiply it by the derivative of the intermediate variable with respect to the input variable.

Product Rule

The product rule is applied when you have two functions multiplied together. The derivative is found by multiplying the derivative of the first function by the second function and adding it to the product of the first function by the derivative of the second function.

Examples

Chain Rule: f(x) = sin(x^2) Solution: Let u = x^2. Then f(x) = sin(u).

df/dx = cos(u) du/dx

= cos(x^2) * 2x

Product Rule: f(x) = x * sin(x) Solution:

df/dx = (d/dx)(x) * sin(x) + x * (d/dx)(sin(x))

= 1 * sin(x) + x * cos(x)

Conclusion

Mastering the chain rule and product rule is crucial for differentiating complex functions. By understanding these techniques, you can effectively analyze and derive the derivatives of functions in various mathematical applications.