Derivative Rules

Initially, let’s discuss the Power Rule. If we have a function of the form f(x) = x^n, who can tell me what the derivative looks like?

Is it something like d/dx[x^n] = n * x^(n-1)?

Exactly! So, if f(x) = x^3, can anyone find f'(x)?

That would be 3x^2!

Great job! A mnemonic to remember this is 'Drop and Multiply', where you drop the exponent and multiply it by the base reduced by one. Let’s move to the next rule.

Now, let’s look at the Sum Rule. When you differentiate a sum of functions, what happens?

We just differentiate each function separately and add the results?

Exactly! d/dx[f(x) + g(x)] = f'(x) + g'(x). Can you apply this to f(x) = x^2 + 3x?

So, f'(x) would be 2x + 3!

Right again! Remember, just differentiate each part separately. Let’s advance to the Product Rule.

Next is the Product Rule. When differentiating a product of two functions, what should we do?

You differentiate the first function and multiply it by the second function, and then add the first function multiplied by the derivative of the second?

Exactly, it’s g'(x)h(x) + g(x)h'(x). Can anyone give me an example?

For f(x) = x^2 * sin(x), f'(x) would be 2x * sin(x) + x^2 * cos(x)!

Correct! Always remember to apply both functions in that formula. Let’s explore the Quotient Rule next.

The Quotient Rule is next! Can someone detail how we differentiate a function that is a quotient?

We use (g'(x)h(x) - g(x)h'(x)) / h(x)^2?

Fantastic! Suppose we have f(x) = x^2/(x + 1). What’s the derivative?

That would be [2x(x + 1) - x^2(1)] / (x + 1)^2!

Exactly! Just remember, it’s crucial to keep the denominator squared in the result.

Lastly, let’s discuss the Chain Rule. What’s the process when differentiating composite functions?

You take the derivative of the outer function and multiply it by the derivative of the inner function?

Correct! d/dx[f(g(x))] = f'(g(x))g'(x). Can anyone provide an example?

For f(x) = sin(x^2), the derivative would be cos(x^2) * 2x!

Excellent! Always make sure to identify both layers of your composite function. Let’s recap what we learned today!