Check again whether you miss any factor in front of ##\Delta x## in the numerator.Karol said:$$=\lim_{\Delta x\rightarrow 0}\frac{\sqrt{2x+3+\Delta x}-\sqrt{2x+3}}{\Delta x}$$
Karol said:Homework Statement
Use the definition of the derivative to find dy/dx for ##~y=\sqrt{2x+3}##
Homework Equations
Derivative as a limit:
$$y'=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$The Attempt at a Solution
$$\lim_{\Delta x\rightarrow 0}\frac{\sqrt{2(x+\Delta x)+3}-\sqrt{2x+3}}{\Delta x}$$
$$=\lim_{\Delta x\rightarrow 0}\frac{\sqrt{2x+3+\Delta x}-\sqrt{2x+3}}{\Delta x}$$
The nominator and denominator tend to 0 but i can't find the limit
$$\lim_{h\rightarrow 0}\sqrt{1+h}=1$$Ray Vickson said:Can you find the behavior of ##\sqrt{1 + h}## or ##\sqrt{1+h} - 1## for small ##h = 2 \Delta x/(2x+3)##?
Karol said:$$\frac{\sqrt{2(x+\Delta x)+3}-\sqrt{2x+3}}{\Delta x}=\frac{\sqrt{2(x+\Delta x)+3}-\sqrt{2x+3}}{\Delta x} \cdot \frac{\sqrt{2(x+\Delta x)+3}+\sqrt{2x+3}}{\sqrt{2(x+\Delta x)+3}+\sqrt{2x+3}}=\frac{2\Delta x}{\Delta x(\sqrt{2(x+\Delta x)+3}+\sqrt{2x+3})}=\frac{1}{\sqrt{2x+3}}$$
$$\lim_{h\rightarrow 0}\sqrt{1+h}=1$$
$$=\lim_{\Delta x\rightarrow 0}\frac{\sqrt{2x+3+2\Delta x}-\sqrt{2x+3}}{\Delta x}=\frac{\sqrt{2x+3}-\sqrt{2x+3}}{\Delta x}=\frac{0}{0}$$
Alternatively:Karol said:$$\frac{\sqrt{2x+3+2\Delta x}-\sqrt{2x+3}}{\Delta x}=\frac{ \sqrt{2x+3} \sqrt{1 + \frac{2 \Delta x}{2x+3}}-\sqrt{2x+3}}{\Delta x}=\frac{\sqrt{2x+3}\left[ \sqrt{1+\frac{2\Delta x}{2x+3}}-1 \right]}{\Delta x}$$
$$=\frac{\sqrt{2x+3}\left[ \sqrt{1+\frac{2\Delta x}{2x+3}}-1 \right]}{\Delta x} \cdot \frac{ \sqrt{1+\frac{2\Delta x}{2x+3}}+1 }{ \sqrt{1+\frac{2\Delta x}{2x+3}}+1 }$$
$$=\frac{1}{\sqrt{2x+3}}$$
A derivative as a limit is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of a tangent line to the graph of the function at that point.
The derivative of a function can be calculated by taking the limit of the difference quotient as the change in the input value approaches zero. This process is known as the limit definition of a derivative.
The limit definition of a derivative is the most fundamental way of calculating derivatives, as it is based on the concept of a limit. Other methods, such as the power rule or product rule, are shortcuts that can be used for certain types of functions.
No, a derivative cannot exist at a discontinuity. This is because the derivative represents the slope of a tangent line, and a discontinuity indicates a sharp change in the slope of a function.
The concept of a derivative is important because it allows us to analyze and understand the behavior of functions. It is used in many areas of mathematics and science, including calculus, physics, and economics. It also has practical applications in fields such as engineering and computer science.