Question

Use the General Power Rule where appropriate to find the derivative of the following functions. $$f(x)=2 x^{\sqrt{2}}$$

Short answer

Answer: The derivative of the function \(f(x)=2x^{\sqrt{2}}\) is \(f'(x)=2\sqrt{2}x^{\sqrt{2}-1}\).

Step-by-step solution

Identify the values of \(a\) and \(n\)

In the given function \(f(x)=2x^{\sqrt{2}}\), we can see that the coefficient \(a=2\) and the exponent \(n=\sqrt{2}\).

Apply the General Power Rule

Now apply the General Power Rule to find the derivative \(f'(x)\): $$f'(x) = an x^{n-1} $$

Substitute the values of \(a\) and \(n\)

Plug in the values of \(a=2\) and \(n=\sqrt{2}\) in the formula: $$f'(x) = 2(\sqrt{2})x^{\sqrt{2}-1}$$

Final Answer

The derivative of the given function is: $$f'(x) = 2\sqrt{2}x^{\sqrt{2}-1}$$

Key concepts

General Power Rule

The General Power Rule is a powerful tool in calculus used to find the derivative of functions of the form \( f(x) = ax^n \), where \( a \) is a constant, and \( n \) is any real number exponent. This rule is a generalization of the simple power rule that you may already know: \( \frac{d}{dx} x^n = nx^{n-1} \). In the case of the General Power Rule, it becomes: - \( f'(x) = anx^{n-1} \)
This means that you multiply the exponent \( n \) by the coefficient \( a \), and then reduce the exponent by 1. This approach allows us to handle more complex functions, including those with fractional or irrational exponents, like \( f(x)=2x^{\sqrt{2}} \). Understanding this rule simplifies the process of differentiation significantly, especially when dealing with non-integer powers.

Exponentiation

Exponentiation involves raising a base number to the power of an exponent, which indicates how many times the base is multiplied by itself. In our example \( f(x)=2x^{\sqrt{2}} \), the exponent is \( \sqrt{2} \). This non-integer exponent might look intimidating but it follows the same principles as integer exponents.
The key points about exponentiation that help in calculus include:

• The basic definition: \( a^n = a \times a \times \ldots \) (n times).
• Non-integer exponents like \( \sqrt{2} \) imply roots and are mathematically valid.
• The rules for exponents (e.g., \( x^a \cdot x^b = x^{a+b} \)) also apply to derivatives.

Exponentiation is essential when applying the General Power Rule, as we adjust the exponent by subtracting one to find the derivative.

Coefficient in functions

In calculus, the coefficient is simply the constant term that multiplies the variable in a function. In our function \( f(x)=2x^{\sqrt{2}} \), the coefficient is \( 2 \).
Understanding coefficients is crucial because:

• The coefficient affects the "rate" at which the function changes; it scales the function.
• When differentiating, the coefficient propagates through; it multiplies the new expression resulting from the differentiation process.
• In the General Power Rule, it is part of what you multiply the exponent by: \( f'(x) = anx^{n-1} \), where \( a \) is the coefficient.

For our function, after applying the General Power Rule, the coefficient \( 2 \) is multiplied by \( \sqrt{2} \) to give the final term in the derivative \( f'(x) = 2\sqrt{2}x^{\sqrt{2}-1} \), emphasizing how crucial the coefficient's role is in shaping the function's slope at any point.