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Chapter 7: Problem 32

Use the General Power Rule to find the derivative of the function. $$ g(x)=\sqrt{5-3 x} $$

Short Answer

Expert verified
Using General Power Rule, the derivative of the function \(g(x)=\sqrt{5-3x}\) is \(g'(x) = -\frac{3}{2\sqrt{5-3x}}\).

Step by step solution

01

Rewrite using Power Rule

First, we write the function using a power instead of a root. Thus, \( g(x) = \sqrt{5-3x} \) is rewritten as \( g(x) = (5-3x)^{\frac{1}{2}} \).
02

Apply Power Rule

Then, apply the Power Rule for differentiation, which states that the derivative of \( u^n \) is \( n \cdot u^{n-1} \cdot u' \). Here, \( u = 5-3x \) and \( n= \frac{1}{2} \). So, \( g'(x) = \frac{1}{2} \cdot (5-3x)^{\frac{1}{2} - 1} \cdot (-3) = - \frac{3}{2} \cdot (5-3x)^{-\frac{1}{2}} \).
03

Simplify The Result

Finally, rewrite the negative exponents and simplify expression : \( g'(x) = -\frac{3}{2} \cdot \frac{1}{\sqrt{5-3x}} \), which simplifies to \( g'(x) = -\frac{3}{2\sqrt{5-3x}} \).

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Most popular questions from this chapter

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=(x+2)^{-1 / 2} $$

Use the General Power Rule to find the derivative of the function. $$ h(t)=\left(1-t^{2}\right)^{4} $$

Use the General Power Rule to find the derivative of the function. $$ f(t)=\sqrt{t+1} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\left(4-3 x^{2}\right)^{-2 / 3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\sqrt{x}(x-2)^{2} $$
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