Math Problem Statement

derivative

If f(x) = 6x√x + 6/(x^2√x), then find f'(4).

Solution

The problem asks for the derivative f(x)f'(x) of the function f(x)=6xx+6x2xf(x) = 6x\sqrt{x} + \frac{6}{x^2\sqrt{x}}, and then to evaluate f(4)f'(4).

Step 1: Simplify the Function

Before differentiating, it will be easier to express the function in terms of powers of xx:

  • Recall that x=x1/2\sqrt{x} = x^{1/2}, so: 6xx=6xx1/2=6x3/26x\sqrt{x} = 6x \cdot x^{1/2} = 6x^{3/2}
  • The second term is 6x2x\frac{6}{x^2\sqrt{x}}. Using x=x1/2\sqrt{x} = x^{1/2}, we have: 6x2x=6x2x1/2=6x5/2\frac{6}{x^2\sqrt{x}} = 6x^{-2} \cdot x^{-1/2} = 6x^{-5/2} So the function becomes: f(x)=6x3/2+6x5/2f(x) = 6x^{3/2} + 6x^{-5/2}

Step 2: Differentiate f(x)f(x)

Now we differentiate term by term using the power rule:

  • The derivative of 6x3/26x^{3/2} is: ddx(6x3/2)=632x1/2=9x1/2\frac{d}{dx}(6x^{3/2}) = 6 \cdot \frac{3}{2} x^{1/2} = 9x^{1/2}
  • The derivative of 6x5/26x^{-5/2} is: ddx(6x5/2)=652x7/2=15x7/2\frac{d}{dx}(6x^{-5/2}) = 6 \cdot \frac{-5}{2} x^{-7/2} = -15x^{-7/2}

So, the derivative of f(x)f(x) is: f(x)=9x1/215x7/2f'(x) = 9x^{1/2} - 15x^{-7/2}

Step 3: Find f(4)f'(4)

Now we substitute x=4x = 4 into the derivative expression: f(4)=941/21547/2f'(4) = 9 \cdot 4^{1/2} - 15 \cdot 4^{-7/2}

First, calculate the powers of 4:

  • 41/2=4=24^{1/2} = \sqrt{4} = 2
  • 47/2=147/2=11284^{-7/2} = \frac{1}{4^{7/2}} = \frac{1}{128}

Now, substitute these values: f(4)=92151128=1815128f'(4) = 9 \cdot 2 - 15 \cdot \frac{1}{128} = 18 - \frac{15}{128}

Next, find a common denominator to simplify: 18=2304128,sof(4)=230412815128=228912818 = \frac{2304}{128}, \quad \text{so} \quad f'(4) = \frac{2304}{128} - \frac{15}{128} = \frac{2289}{128}

Thus, f(4)=2289128f'(4) = \frac{2289}{128}.

Final Answer:

f(4)=2289128f'(4) = \frac{2289}{128}.

Would you like more details or have any questions?

Here are five related questions:

  1. How do you differentiate functions with fractional exponents?
  2. What are some common applications of the power rule in real-world problems?
  3. How does the chain rule apply to expressions involving square roots?
  4. What is the difference between differentiating polynomials and rational functions?
  5. How can you simplify complex functions before applying derivative rules?

Tip: Always simplify expressions before differentiating, as it makes the process easier and reduces the chance of errors.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Power Rule

Formulas

Power Rule: d/dx[x^n] = n*x^(n-1)
Derivative of a constant multiple

Theorems

Power Rule

Suitable Grade Level

Grades 11-12