Math Problem Statement

question number 6

Differentiate the following functions, giving your answer in a similar form, without negative or fractional indices: (a) f(x) = 1/x (b) f(x) = √x (c) f(x) = 1/√x (d) y = x√x

Solution

Let's focus on Question 6 from the provided image, which involves differentiating various functions.

The given functions are:

  • (a) f(x)=1xf(x) = \frac{1}{x}
  • (b) f(x)=xf(x) = \sqrt{x}
  • (c) f(x)=1xf(x) = \frac{1}{\sqrt{x}}
  • (d) y=xxy = x \sqrt{x}

We will differentiate each of these step by step.


(a) f(x)=1xf(x) = \frac{1}{x}

We rewrite the function using a negative exponent: f(x)=x1f(x) = x^{-1}

Using the power rule (ddx[xn]=nxn1\frac{d}{dx}[x^n] = n \cdot x^{n-1}), we get: f(x)=1x2=1x2f'(x) = -1 \cdot x^{-2} = -\frac{1}{x^2}


(b) f(x)=xf(x) = \sqrt{x}

We rewrite the square root as a fractional exponent: f(x)=x12f(x) = x^{\frac{1}{2}}

Using the power rule: f(x)=12x12=12xf'(x) = \frac{1}{2} \cdot x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}


(c) f(x)=1xf(x) = \frac{1}{\sqrt{x}}

We rewrite the function using a negative fractional exponent: f(x)=x12f(x) = x^{-\frac{1}{2}}

Using the power rule: f(x)=12x32=12x32=12x3f'(x) = -\frac{1}{2} \cdot x^{-\frac{3}{2}} = -\frac{1}{2x^{\frac{3}{2}}} = -\frac{1}{2\sqrt{x^3}}


(d) y=xxy = x \sqrt{x}

We rewrite the function: y=xx12y = x \cdot x^{\frac{1}{2}}

Using the laws of exponents (aman=am+na^m \cdot a^n = a^{m+n}): y=x1+12=x32y = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}}

Now differentiate using the power rule: y=32x12=3x2y' = \frac{3}{2} \cdot x^{\frac{1}{2}} = \frac{3\sqrt{x}}{2}


Final Answers:

  1. (a) f(x)=1x2f'(x) = -\frac{1}{x^2}
  2. (b) f(x)=12xf'(x) = \frac{1}{2\sqrt{x}}
  3. (c) f(x)=12x3f'(x) = -\frac{1}{2\sqrt{x^3}}
  4. (d) y=3x2y' = \frac{3\sqrt{x}}{2}

Would you like a more detailed explanation of any specific part? Let me know!


Here are 5 related questions for practice:

  1. Differentiate f(x)=1x2f(x) = \frac{1}{x^2}.
  2. Differentiate f(x)=x3xf(x) = x^3 \sqrt{x}.
  3. Differentiate f(x)=1x3f(x) = \frac{1}{x^3}.
  4. Differentiate f(x)=x5f(x) = \sqrt{x^5}.
  5. Differentiate f(x)=x3/2f(x) = x^{-3/2}.

Tip: Always rewrite roots and fractions using exponents before applying differentiation rules—it simplifies the process.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Power Rule
Exponent Rules

Formulas

Power Rule: d/dx[x^n] = n * x^(n-1)
Laws of Exponents: a^m * a^n = a^(m+n), 1/a^n = a^(-n)

Theorems

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Suitable Grade Level

Grades 10-12