Category Archives: Derivative

Lagrange Multipliers

1. So, last time we considered finding extrema for some function , but what if we constrain our focus to some surface ? For example, the unit circle would have (1) How do we find extrema for on the unit … Continue reading

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Directional Derivative, Gradient

1. Suppose we have a scalar function of several variables (1) Let be some unit vector. How does change in the direction? We can consider this quantity as a function (2) What does this look like?

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Chain Rule for Partial Derivatives

1. Problem. Consider a function where we parametrize (1) If , how does change? We will need to use partial derivatives and Taylor expansion…

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Continuity for Functions of Several Variables, Partial Derivatives

1. We considered differentiating and integrating functions of a single-variable. How? We began with the notion of a limit, and then considered the derivative. If we have a, e.g., polynomial (1) we see (2) Again we stop and reflect: this … Continue reading

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L’Hopital’s Rule for Limits

1. Now we are concerned about limits and we can have nightmarish limits. For example if (1a) then (1b)? What can we do? We end up using derivatives to help evaluate such limits. We have L’Hopital’s rule state, if , … Continue reading

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Inverse Function Theorem

1. Introduction. So we have some function . The “Inverse Function” to is another function denoted satisfying (1). WARNING: do not confuse with . Observe we have be a function of . So what happens if we differentiate both sides … Continue reading

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Differentiating Trigonometric Functions

1. Introduction. So we have investigated several useful trig limits, lets consider the bigger problem: differentiating trigonometric functions. We will use the following properties without proving them: (1a) (1b) (1c) (1d) Where we have be arbitrary real numbers. 2. Derivative … Continue reading

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