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1 year ago

Pls Help - Calc. HW dy/dx problem

Mathematics

1 answer:

GREYUIT [131]1 year ago

8 0

Answer:

$\displaystyle y' = \frac{-2}{x \ln (10)[\log (x) - 2]^2}$

General Formulas and Concepts:

Calculus

Differentiation

Derivatives
Derivative Notation

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Derivative Rule [Basic Power Rule]:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]: $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle y = \frac{\log (x)}{\log (x) - 2}$

Step 2: Differentiate

[Function] Derivative Rule [Quotient Rule]: $\displaystyle y' = \frac{[\log (x) - 2][\log (x)]' - [\log (x) - 2]'[\log (x)]}{[\log (x) - 2]^2}$
Rewrite [Derivative Rule - Addition/Subtraction]: $\displaystyle y' = \frac{[\log (x) - 2][\log (x)]' - [\log (x)' - 2'][\log (x)]}{[\log (x) - 2]^2}$
Logarithmic Differentiation: $\displaystyle y' = \frac{[\log (x) - 2]\frac{1}{\ln (10)x} - [\frac{1}{\ln (10)x} - 2'][\log (x)]}{[\log (x) - 2]^2}$
Derivative Rule [Basic Power Rule]: $\displaystyle y' = \frac{[\log (x) - 2]\frac{1}{\ln (10)x} - \frac{1}{\ln (10)x}[\log (x)]}{[\log (x) - 2]^2}$
Simplify: $\displaystyle y' = \frac{\frac{\log (x) - 2}{\ln (10)x} - \frac{\log (x)}{\ln (10)x}}{[\log (x) - 2]^2}$
Simplify: $\displaystyle y' = \frac{\frac{-2}{\ln (10)x}}{[\log (x) - 2]^2}$
Rewrite: $\displaystyle y' = \frac{-2}{x \ln (10)[\log (x) - 2]^2}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

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Answer:

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2 years ago

Lim (n/3n-1)^(n-1) n → ∞

n200080 [17]

Looks like the given limit is

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With some simple algebra, we can rewrite

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For the second limit, recall the definition of the constant, e :

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So, the overall limit is indeed 0:

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