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:

ddxcf(x)=cf(x)ddx(f(x)+g(x))=ddxf(x)+ddxg(x)ddxc=0f(x)=u(x)v(x)f(x)=u(x)v(x)+v(x)u(x)\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:

ddxxn=nxn1ddxln(x)=1xddxex=exddxsin(x)=cos(x)ddxcos(x)=sin(x)\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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