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Chain Rule: Advanced Derivative Examples
The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function. A composite function is a function within a function, like f(g(x)). The chain rule states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.
In simpler terms, if we have a function y = f(u) and another function u = g(x), then the chain rule states that the derivative of y with respect to x is:
dy/dx = dy/du * du/dx
This might seem complicated at first, but it becomes easier with practice. Let's look at some advanced examples to illustrate the application of the chain rule.
Example 1: Finding the Derivative of a Trigonometric Function
Let's find the derivative of the function y = sin(x^2). Here, we have a composite function where the outer function is sin(u) and the inner function is u = x^2.
Using the chain rule, we get:
dy/dx = dy/du * du/dx
dy/du = cos(u) (derivative of the outer function)
du/dx = 2x (derivative of the inner function)
Substituting these values, we get:
dy/dx = cos(u) * 2x
Since u = x^2, we can substitute back to get:
dy/dx = cos(x^2) * 2x
Example 2: Finding the Derivative of an Exponential Function
Let's find the derivative of the function y = e^(3x + 1). Here, the outer function is e^u and the inner function is u = 3x + 1.
Using the chain rule, we get:
dy/dx = dy/du * du/dx
dy/du = e^u (derivative of the outer function)
du/dx = 3 (derivative of the inner function)
Substituting these values, we get:
dy/dx = e^u * 3
Since u = 3x + 1, we can substitute back to get:
dy/dx = 3e^(3x + 1)
Example 3: Finding the Derivative of a Function with Multiple Composite Functions
Let's find the derivative of the function y = (2x^2 + 1)^3. Here, we have two composite functions: the outer function u^3 and the inner function u = 2x^2 + 1.
Using the chain rule twice, we get:
dy/dx = dy/du * du/dx
dy/du = 3u^2 (derivative of the outer function)
du/dx = 4x (derivative of the inner function)
Substituting these values, we get:
dy/dx = 3u^2 * 4x
Since u = 2x^2 + 1, we can substitute back to get:
dy/dx = 3(2x^2 + 1)^2 * 4x
dy/dx = 12x(2x^2 + 1)^2
Conclusion
The chain rule is a powerful tool in calculus that allows us to find derivatives of complex functions. By understanding the chain rule and practicing its application, you can confidently tackle more advanced derivative problems.