A Comprehensive Guide on How to Find Derivatives: Power Rule, Product Rule, Quotient Rule, and Chain Rule Explained

f'(x)

To find the derivative of a function f(x), we use differentiation rules

To find the derivative of a function f(x), we use differentiation rules. The derivative, denoted by f'(x) or dy/dx, gives us the rate of change of the function at any point x.

To differentiate a function, we need to apply the power rule, product rule, quotient rule, and chain rule as necessary. The specific steps will depend on the form of the function.

For example, if we have the function f(x) = x^2 + 3x – 2, we can find its derivative f'(x) as follows:

Step 1: Apply the power rule to each term separately.
– The derivative of x^2 is 2x.
– The derivative of 3x is 3.

Step 2: Combine the derivatives of each term.
– The derivative of x^2 + 3x – 2 is 2x + 3.

So f'(x) = 2x + 3.

Another example could be a composite function such as f(x) = (2x + 1)^4. In this case, we can use the chain rule:

Step 1: Identify the inner function: 2x + 1.
Step 2: Find the derivative of the inner function: (d/dx)(2x + 1) = 2.
Step 3: Keep the outer function as it is and multiply by the derivative of the inner function.
– f'(x) = 4(2x + 1)^3 * 2 = 8(2x + 1)^3.

Therefore, the derivative of f(x) = (2x + 1)^4 is f'(x) = 8(2x + 1)^3.

These are just two examples of finding derivatives, and the steps may vary depending on the specific function. It’s important to understand the rules of differentiation and practice solving different types of functions to become familiar with the process.

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