# Derivatives

Function:

x = 1

Show Tangent Line at x

First Derivative

Complete

Second Derivative

Complete

x = 1

Show Tangent Line at x

First Derivative

Complete

Second Derivative

Complete

Function:

x = 1

Show Tangent Line at x

First Derivative

Complete

Second Derivative

Complete

x = 1

Show Tangent Line at x

First Derivative

Complete

Second Derivative

Complete

The first derivative \(f'(x)\) of a function \(f(x)\) is defined as the limit
\[ \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\]
At a particular value \(a\), we can also define \(f'(a)\) as
\[ f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\]
if this limit exists. This limit will be equal to the slope of the tangent line to \(f(x)\) at \(x=a\).

The second derivative \(f''(x)\) is found by taking the derivative of the first derivative. That is, the value of \(f''(x)\) is the slope of the tangent line to the graph of \(f'(x)\) at \(x\).

The second derivative \(f''(x)\) is found by taking the derivative of the first derivative. That is, the value of \(f''(x)\) is the slope of the tangent line to the graph of \(f'(x)\) at \(x\).

Choose a function to work with, and use the slider to adjust the current value of \(x\). The tangent line to \(f(x)\) at \(x\) will be displayed, along with the graphs of \(f'(x)\) and \(f''(x)\) if these options are enabled.

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