Find the derivative of the following: 1. √x / (x + 1) 2. (x² + 1)(x³ - 1) 3. (ax + b) / (cx + d) 4. (x - 3)(2x + 1)(x + 5) 5. Suppose f(2) = 3, g(2) = 2, f'(2) = -2, g'(2) = 4. Fin... Find the derivative of the following: 1. √x / (x + 1) 2. (x² + 1)(x³ - 1) 3. (ax + b) / (cx + d) 4. (x - 3)(2x + 1)(x + 5) 5. Suppose f(2) = 3, g(2) = 2, f'(2) = -2, g'(2) = 4. Find h'(2) where h(x) = 5f(x) + 2g(x), h(x) = f(x)g(x), h(x) = g(x)/f(x).
Understand the Problem
The question involves calculating the derivatives of several mathematical expressions and utilizing the chain rule and product rule for differentiation. Each expression requires applying appropriate derivative rules to find the result.
Answer
1. $y' = \frac{\frac{\sqrt{x}}{2\sqrt{x}}(x + 1) - \sqrt{x}(1)}{(x + 1)^2}$ 2. $y' = (2x)(x^3 - 1) + (x^2 + 1)(3x^2)$ 3. $y' = \frac{a(cx + d) - (ax + b)c}{(cx + d)^2}$ 4. $y' = (1)(2x + 1)(x + 5) + (x - 3)(2)(x + 5) + (x - 3)(2x + 1)(1)$ 5. $h'(2) = -2, \, 8, \, \frac{16}{9}$
Answer for screen readers
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$y' = \frac{\frac{\sqrt{x}}{2\sqrt{x}}(x + 1) - \sqrt{x}(1)}{(x + 1)^2}$
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$y' = (2x)(x^3 - 1) + (x^2 + 1)(3x^2)$
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$y' = \frac{a(cx + d) - (ax + b)c}{(cx + d)^2}$
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$y' = (1)(2x + 1)(x + 5) + (x - 3)(2)(x + 5) + (x - 3)(2x + 1)(1)$
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$h'(2) = -2, , 8, , \frac{16}{9}$
Steps to Solve
- Derivative of $\frac{\sqrt{x}}{x + 1}$ using Quotient Rule
For the quotient rule, if $y = \frac{u}{v}$, then the derivative $y' = \frac{u'v - uv'}{v^2}$.
Here, let $u = \sqrt{x}$ and $v = x + 1$.
Calculating $u'$ and $v'$:
- $u' = \frac{1}{2\sqrt{x}}$
- $v' = 1$
Now applying the quotient rule: $$ y' = \frac{\left(\frac{1}{2\sqrt{x}}\right)(x + 1) - \sqrt{x}(1)}{(x + 1)^2} $$
- Derivative of $(x^2 + 1)(x^3 - 1)$ using Product Rule
For the product rule, if $y = uv$, then $y' = u'v + uv'$.
Let $u = x^2 + 1$ and $v = x^3 - 1$.
Calculating $u'$ and $v'$:
- $u' = 2x$
- $v' = 3x^2$
Now applying the product rule: $$ y' = (2x)(x^3 - 1) + (x^2 + 1)(3x^2) $$
- Derivative of $\frac{ax + b}{cx + d}$ using Quotient Rule
Let $u = ax + b$ and $v = cx + d$.
Calculating $u'$ and $v'$:
- $u' = a$
- $v' = c$
Applying the quotient rule: $$ y' = \frac{a(cx + d) - (ax + b)c}{(cx + d)^2} $$
- Derivative of $(x - 3)(2x + 1)(x + 5)$ using Product Rule
Let $u = (x - 3)$, $v = (2x + 1)$, and $w = (x + 5)$.
Using the product rule for three functions: $$ y' = u'vw + uv'w + uvw' $$
Calculating derivatives:
- $u' = 1$, $v' = 2$, $w' = 1$
Now applying: $$ y' = (1)(2x + 1)(x + 5) + (x - 3)(2)(x + 5) + (x - 3)(2x + 1)(1) $$
- Find $h'(2)$ for $h(x) = 5f(x) + 2g(x)$
Using the derivative rule: $$ h' = 5f'(x) + 2g'(x) $$
Substituting $x = 2$: $$ h'(2) = 5(-2) + 2(4) = -10 + 8 = -2 $$
- Find $h'(2)$ for $h(x) = f(x)g(x)$
Using the product rule: $$ h' = f'(x)g(x) + f(x)g'(x) $$
Substituting $x = 2$: $$ h'(2) = (-2)(2) + (3)(4) = -4 + 12 = 8 $$
- Find $h'(2)$ for $h(x) = \frac{g(x)}{f(x)}$
Using the quotient rule: $$ h' = \frac{g'(x)f(x) - g(x)f'(x)}{(f(x))^2} $$
Substituting $x = 2$: $$ h'(2) = \frac{(4)(3) - (2)(-2)}{(3)^2} = \frac{12 + 4}{9} = \frac{16}{9} $$
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$y' = \frac{\frac{\sqrt{x}}{2\sqrt{x}}(x + 1) - \sqrt{x}(1)}{(x + 1)^2}$
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$y' = (2x)(x^3 - 1) + (x^2 + 1)(3x^2)$
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$y' = \frac{a(cx + d) - (ax + b)c}{(cx + d)^2}$
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$y' = (1)(2x + 1)(x + 5) + (x - 3)(2)(x + 5) + (x - 3)(2x + 1)(1)$
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$h'(2) = -2, , 8, , \frac{16}{9}$
More Information
The derivatives found illustrate the application of different rules of differentiation—such as the product rule and quotient rule—across various mathematical expressions. Understanding these concepts is essential for more advanced calculus applications.
Tips
- Forgetting to apply the product or quotient rules correctly.
- Not simplifying expressions fully after taking derivatives.
- Mixing up signs in the derivatives, especially when using negative derivatives.
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