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Chapter 3: Problem 36

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

Short Answer

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

Step by step solution

01

Identify the constants and variables

In the given function $$f(x) = 2x^{\sqrt{2}}$$, $$2$$ is the constant multiplier and the exponent root$$2$$ is the power of the variable $$x$$.
02

Apply the General Power Rule

Use the General Power Rule to find the derivative, $$f'(x)$$: $$f'(x) = n \cdot x^{n-1} \cdot g'(x)$$ In this case, n is the power $$\sqrt{2}$$, and $$g'(x)$$ denotes the derivative of the inner function $$g(x) = x$$, which is simply $$1$$.
03

Calculate the expression

Plug in the values of the constants and variables into the expression: $$f'(x) = \sqrt{2} \cdot x^{\sqrt{2}-1} \cdot 1$$
04

Simplify the expression

The expression simplifies to: $$f'(x) = \sqrt{2} \cdot x^{\sqrt{2}-1}$$
05

Multiply by the constant

Now, multiply the derivative by the constant $$2$$: $$f'(x) = 2 \cdot \sqrt{2} \cdot x^{\sqrt{2}-1}$$
06

Write the final solution

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

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