Math Problem Statement

take the derivative of y=9/square root of (x+7)

Solution

To find the derivative of y=9x+7y = \frac{9}{\sqrt{x+7}}, let's first rewrite it in a form that makes it easier to differentiate:

y=9(x+7)12y = 9(x+7)^{-\frac{1}{2}}

Now, we'll use the power rule for derivatives, which states that if f(x)=xnf(x) = x^n, then f(x)=nxn1f'(x) = n \cdot x^{n-1}.

Let's differentiate:

= 9 \cdot \left(-\frac{1}{2}\right) \cdot (x+7)^{-\frac{3}{2}} \cdot \frac{d}{dx}(x+7)$$ Since $$\frac{d}{dx}(x+7) = 1$$, the derivative simplifies to: $$y' = -\frac{9}{2} (x+7)^{-\frac{3}{2}}$$ Finally, let's rewrite the result: $$y' = -\frac{9}{2\sqrt{(x+7)^3}}$$ So, the derivative of $$y = \frac{9}{\sqrt{x+7}}$$ is: $$y' = -\frac{9}{2\sqrt{(x+7)^3}}$$ --- Would you like further details on any of the steps? Or do you have another question? Here are some related questions to explore: 1. How do you differentiate a function with a fractional exponent? 2. What is the chain rule and how does it apply to this derivative? 3. How do you handle derivatives involving square roots or radicals? 4. Can you find the second derivative of $$y$$? 5. How would you find the derivative of a rational function? **Tip**: When dealing with expressions involving square roots, it's often helpful to rewrite them with fractional exponents to make differentiation easier.

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Math Problem Analysis

Mathematical Concepts

Derivatives
Power Rule
Chain Rule
Fractional Exponents

Formulas

Power rule for derivatives: f'(x) = n * x^(n-1)
Chain rule
Rewriting square roots as fractional exponents

Theorems

Power Rule
Chain Rule

Suitable Grade Level

Grades 11-12