Chain Rule
The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function
The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function. It is commonly used when we have functions within other functions, such as trigonometric functions inside logarithmic functions, or polynomials inside exponential functions.
To understand the chain rule, let’s consider a composite function, f(g(x)). The chain rule states that the derivative of this composite function can be found by multiplying the derivative of the outer function, f'(g(x)), by the derivative of the inner function, g'(x).
In mathematical notation, the chain rule can be expressed as: (f(g(x)))’ = f'(g(x)) * g'(x).
To apply the chain rule, follow these steps:
Step 1: Identify the outer function, which is usually the function that contains the other function.
Step 2: Identify the inner function, which is the function embedded within the outer function.
Step 3: Find the derivative of the outer function, f'(g(x)), treating the inner function, g(x), as a variable.
Step 4: Find the derivative of the inner function, g'(x).
Step 5: Multiply f'(g(x)) by g'(x) to get the derivative of the composite function.
Let’s work through an example to illustrate the use of the chain rule:
Find the derivative of f(x) = sin(2x^2 + 3x – 1).
In this example, the outer function is sin(x), while the inner function is 2x^2 + 3x – 1.
Step 1: The outer function is sin(x).
Step 2: The inner function is 2x^2 + 3x – 1.
Step 3: Find the derivative of the outer function: f'(x) = cos(x).
Step 4: Find the derivative of the inner function: g'(x) = 4x + 3.
Step 5: Multiply f'(g(x)) by g'(x): f'(g(x)) * g'(x) = cos(2x^2 + 3x – 1) * (4x + 3).
Therefore, the derivative of f(x) = sin(2x^2 + 3x – 1) is cos(2x^2 + 3x – 1) * (4x + 3).
By applying the chain rule, we can find the derivative of composite functions, which can be extremely helpful in various mathematical applications, particularly in calculus and physics.
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