Compute the directional derivative of f at the given point...

Compute the directional derivative of $f$ at the given point in the direction of the indicated vector.
$$f(x, y)=e^{4 x^{2}-y},(1,4), \text { $\mathbf{u}$ in the direction of }\langle-2,-1\rangle$$

Key Concepts

Directional Derivative

The directional derivative measures the rate at which a function changes at a given point in the direction of a specified vector. It quantifies the slope of the function in that particular direction and is an extension of the concept of a derivative from one-dimensional calculus to higher dimensions.

Gradient

The gradient of a function is a vector that consists of all its first-order partial derivatives. It points in the direction of the steepest ascent of the function, and its magnitude indicates the rate of increase in that direction. The gradient is essential in computing the directional derivative since its dot product with the unit vector in the desired direction gives the derivative.

Unit Vector

A unit vector is a vector of length one that is used to indicate direction. In the context of directional derivatives, the given direction vector is often normalized to ensure that only the directional effect is measured, independent of the vector's magnitude.

Partial Derivative

Partial derivatives involve differentiating a multivariable function with respect to one variable at a time while keeping the other variables constant. They are the building blocks of the gradient vector and are crucial for understanding how a function changes with respect to each of its variables.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number. In the context of the directional derivative, the dot product of the gradient vector with a unit vector in a specified direction yields the rate of change of the function along that direction.

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