Math Problem Statement

What is the derivative of f(x) = √x * e^(6x)?

Solution

We are asked to find the derivative of the function f(x)=xe6xf(x) = \sqrt{x} \cdot e^{6x}.

Step-by-Step Solution:

The function can be rewritten as: f(x)=x1/2e6xf(x) = x^{1/2} \cdot e^{6x}

We'll apply the product rule for differentiation, which states: (fg)=fg+fg(f \cdot g)' = f' \cdot g + f \cdot g' Here, f(x)=x1/2f(x) = x^{1/2} and g(x)=e6xg(x) = e^{6x}.

Step 1: Differentiate x1/2x^{1/2}

The derivative of x1/2x^{1/2} is: ddx(x1/2)=12x1/2=12x\frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}

Step 2: Differentiate e6xe^{6x}

The derivative of e6xe^{6x} is: ddx(e6x)=6e6x\frac{d}{dx} \left( e^{6x} \right) = 6 e^{6x}

Step 3: Apply the product rule

Now, apply the product rule: f(x)=(12xe6x)+(x1/26e6x)f'(x) = \left( \frac{1}{2\sqrt{x}} \cdot e^{6x} \right) + \left( x^{1/2} \cdot 6 e^{6x} \right) Simplifying both terms:

f(x)=e6x2x+6xe6xf'(x) = \frac{e^{6x}}{2\sqrt{x}} + 6\sqrt{x} e^{6x}

Step 4: Combine terms

Factor out e6xe^{6x}: f(x)=e6x(12x+6x)f'(x) = e^{6x} \left( \frac{1}{2\sqrt{x}} + 6\sqrt{x} \right)

Now, express everything under a common denominator: f(x)=e6x(1+12x2x)f'(x) = e^{6x} \left( \frac{1 + 12x}{2\sqrt{x}} \right)

This matches the option: (6x+1)e6x/2x\boxed{(6 \cdot x + 1) \cdot e^{6x} / 2\sqrt{x}}

This corresponds to the second option in the image.


Would you like further clarification or any additional details on this problem?

Here are some related questions to explore:

  1. What is the product rule in calculus, and when do we use it?
  2. How do we differentiate exponential functions involving constants?
  3. What is the chain rule, and how would it apply if the exponent in e6xe^{6x} were more complex?
  4. How can we simplify expressions involving radicals and exponents for derivatives?
  5. Why is factoring out common terms, like e6xe^{6x}, helpful in derivative expressions?

Tip: When applying the product rule, always simplify intermediate results as much as possible before moving on to the next step!

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Math Problem Analysis

Mathematical Concepts

Product Rule
Exponential Functions
Differentiation

Formulas

Product Rule: (f * g)' = f' * g + f * g'
Power Rule: d/dx(x^n) = n * x^(n-1)
Exponential Derivative: d/dx(e^(kx)) = k * e^(kx)

Theorems

Product Rule
Power Rule
Exponential Function Derivative

Suitable Grade Level

Grades 11-12 / Early University Calculus