Solve the calculus problems as presented in the image.
Understand the Problem
The image presents calculus problems, specifically focusing on differentiation. It includes an example of applying the difference rule and finding the derivative of a function involving square roots and negative exponents. It also touches upon finding the equation of the normal to a graph, which involves finding derivatives and reciprocals.
Answer
$\frac{3}{2\sqrt{x}}$
Answer for screen readers
$\frac{3}{2\sqrt{x}}$
Steps to Solve
- Rewrite the square root as a power
We need to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$
So $3\sqrt{x} = 3x^{\frac{1}{2}}$
- Apply the power rule
The power rule states that if $f(x) = ax^n$, then $f'(x) = n \cdot ax^{n-1}$
Applying this rule to $3x^{\frac{1}{2}}$ we get:
$\frac{d}{dx}(3x^{\frac{1}{2}}) = \frac{1}{2} \cdot 3x^{\frac{1}{2} - 1}$
- Simplify the derivative
Simplify the expression to obtain the final derivative:
$\frac{3}{2}x^{-\frac{1}{2}} = \frac{3}{2} \cdot \frac{1}{x^{\frac{1}{2}}} = \frac{3}{2\sqrt{x}}$
$\frac{3}{2\sqrt{x}}$
More Information
The derivative of $3\sqrt{x}$ is $\frac{3}{2\sqrt{x}}$, which represents the instantaneous rate of change of the function $3\sqrt{x}$ with respect to $x$.
Tips
A common mistake is forgetting to apply the power rule correctly, especially when dealing with fractional exponents. Remember to multiply by the exponent and subtract 1 from the exponent. Another mistake is not simplifying the expression completely, leaving the negative or fractional exponent unaddressed.
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