Learn

Congrats! You have finished your exploration of differential calculus. This has been a jam-packed lesson, so pat yourself on the back for making it through the material. Let’s review what we have learned throughout the exercises:

#### Important Concepts

- A
*limit*is the value of a function approaches as we move to some*x*value. - The
*limit definition of a derivative*shows us how to measure the instantaneous rate of change of a function at a specific point. - A
*derivative*is the slope of a tangent line at a specific point. The derivative of a function*f(x)*is denoted as*f’(x)*. - The derivative of a function allows us to determine when a function is increasing, decreasing, or is at a minimum or maximum value.

#### Derivative Rules

We learned the following general derivative rules:

```
$\begin{aligned}
\frac{d}{dx} c f(x) = c f'(x) \\
\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) \\
\frac{d}{dx}c = 0 \\
f(x) = u(x)v(x) \rightarrow f'(x) =u(x)v'(x) + v(x)u'(x)
\end{aligned}$
```

We also learned how to calculate derivatives for explicit functions:

```
$\begin{aligned}
\frac{d}{dx}x^{n} = nx^{n-1}
\frac{d}{dx}ln(x) = \frac{1}{x} \\
\frac{d}{dx}e^x = e^x \\
\frac{d}{dx}sin(x) = cos(x) \\
\frac{d}{dx}cos(x) = -sin(x)
\end{aligned}$
```

#### Derivatives in Python

`np.gradient()`

allows us to calculate derivatives of functions represented by arrays.

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