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Gwapako
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Help with Derivatives Problem - Get Answers Now
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In summary, the conversation involves discussing the limit formula for finding the derivative of a function, where $\Delta x$ is substituted as $h$ for ease of typing. The process involves algebraic manipulation and using properties of limits to simplify the equation. The second limit also requires rationalizing the numerator.
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skeeter
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I assume you’re referring to a form of the following limit ...
$\displaystyle f’(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x}$
for ease in typing it out, let $\Delta x = h$
$\displaystyle y’ = \lim_{h \to 0} \dfrac{1}{h} \left[\dfrac{2(x+h)-1}{2(x+h)+1} - \dfrac{2x-1}{2x+1} \right]$
from here, it’s just the algebra drill of combining the two fractions using a common denominator and ultimately getting the $h$ in the leading $\dfrac{1}{h}$ factor to divide out with an $h$ factor in the numerator.by the same token, let $\Delta t = h$ in the second limit; also, using the properties of limits will make the overall task a bit easier ...
$\displaystyle x’ = 3 \cdot \lim_{h \to 0} \dfrac{(t+h)^2-t^2}{h} - 2 \cdot \lim_{h \to 0} \dfrac{\sqrt{t+h} - \sqrt{t}}{h}$
the limit of the first term is straightforward; the second will require you to rationalize the numerator
$\displaystyle f’(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x}$
for ease in typing it out, let $\Delta x = h$
$\displaystyle y’ = \lim_{h \to 0} \dfrac{1}{h} \left[\dfrac{2(x+h)-1}{2(x+h)+1} - \dfrac{2x-1}{2x+1} \right]$
from here, it’s just the algebra drill of combining the two fractions using a common denominator and ultimately getting the $h$ in the leading $\dfrac{1}{h}$ factor to divide out with an $h$ factor in the numerator.by the same token, let $\Delta t = h$ in the second limit; also, using the properties of limits will make the overall task a bit easier ...
$\displaystyle x’ = 3 \cdot \lim_{h \to 0} \dfrac{(t+h)^2-t^2}{h} - 2 \cdot \lim_{h \to 0} \dfrac{\sqrt{t+h} - \sqrt{t}}{h}$
the limit of the first term is straightforward; the second will require you to rationalize the numerator
1. What is a derivative?
A derivative is a mathematical concept that represents the rate of change of a function with respect to its input variable. It measures how much a function changes as its input variable changes.
2. Why are derivatives important?
Derivatives are important because they have a wide range of applications in mathematics, science, and engineering. They are used to calculate rates of change, find maximum and minimum values, and solve optimization problems.
3. How do I solve a derivative problem?
To solve a derivative problem, you need to use the rules of differentiation, which involve taking the derivative of each term in the function and simplifying the result. You may also need to use the chain rule, product rule, or quotient rule depending on the complexity of the function.
4. Can I use derivatives in real-life situations?
Yes, derivatives are used in many real-life situations, such as calculating the speed of a moving object, determining the rate of change of a stock price, or finding the optimal production level for a company.
5. Are there any resources available to help with derivatives problems?
Yes, there are many resources available to help with derivatives problems, including textbooks, online tutorials, and math tutoring services. You can also use software programs or calculators to solve derivative problems quickly and accurately.
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