91Ó°ÊÓ

The power rule is a fundamental tool in calculus differentiation used to find derivatives of functions of the form \[x^n\] where \(n\) can be any real number. To apply the power rule effectively, follow these simple steps:

• Identify the exponent \(n\).
• Bring down the exponent \(n\) just in front of the function.
• Subtract one from the exponent \(n\).
• Perform any additional simplifications if needed.

In the original exercise, the power rule is used to differentiate \(x^{9/4}\). Applying it, \(n\) is \(9/4\), so we bring down \(9/4\) and then reduce the exponent by one, resulting in \(x^{5/4}\). These simple steps yield: \[\frac{d}{dx}[x^{9/4}] = \frac{9}{4} x^{\frac{5}{4}}\] The power rule makes it efficient to handle polynomial terms, allowing for quick and accurate differentiation.

The chain rule is a crucial technique in calculus used to differentiate compositions of functions. This is particularly useful when dealing with exponential functions where the exponent itself is a function of \(x\). The general form of the chain rule is:

\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]
To apply the chain rule accurately, follow these steps:

• Identify the inner function, \(g(x)\).
• Determine the derivative of the outer function, \(f'(g(x))\), keeping \(g(x)\) as the variable.
• Find the derivative of the inner function, \(g'(x)\).
• Multiply the derivatives together.

In our example, for \(e^{-2x}\), the inner function is \(-2x\). So, the chain rule tells us to multiply the derivative of \(e^{-2x}\), which is \(e^{-2x}\) itself, by the derivative of the inner function, \(g'(x) = -2\). Hence, the derivative is \[-2e^{-2x}\]. The chain rule allows for seamless differentiation of layers of functions.

Exponential functions are expressed in the form \(e^{u(x)}\), where \(e\) is the base of natural logarithms, approximated as 2.718. They are prevalent in calculus due to their unique property where the derivative of an exponential function with a linear exponent can be neatly derived using the chain rule.

Here’s how you deal with differentiating exponential functions:

• Identify the exponent \(u(x)\).
• Apply the exponential derivative \(e^{u(x)}\).
• Multiply by the derivative of the inside function \(u(x)\).

In the exercise, the exponential function is \(e^{-2x}\). Recognizing \(-2x\) as \(u(x)\), its derivative \(-2\) plays a key role in finding: \[-2e^{-2x}\]. The beauty of exponential functions lies in their predictable behavior during differentiation, making them manageable and straightforward with practice.