 Hello, and welcome to this screencast on Section 9.7, Derivatives and Integrals of Vector-Valued Functions. This screencast is going to cover the derivative, computing derivatives, and tangent lines. In Calculus 1, you are introduced to the limit definition of a derivative. This definition extends naturally to vector-valued functions and curves in space, as the limit of the average rate of change of the function, as we have displayed here. Note that we also use the familiar Leibniz notation for the derivative of R. Let's look at a picture. Here we have two vectors, R of t plus h in green and R of t in blue. The quotient in the limit gives us the average rate of change of the function, R, on the interval from t to t plus h. This is denoted by the pink vector in the picture. As h approaches zero, this average rate of change approaches the instantaneous rate of change of R at the point t. The instantaneous rate of change of R at the point t is the derivative of R at t. Just as the limit definition extends naturally to vector-valued functions, so does computing derivatives. To compute the derivative of a vector-valued function R with components x of t, y of t, and z of t, we take the derivative of each component separately with respect to t. The derivative of R then equals a new vector, r prime of t, with components x prime of t, y prime of t, and z prime of t. All of the derivative rules from Calculus 1 can be applied when taking derivatives of the components of R. There are also several analogous rules for vector-valued functions, including a product rule for scalar functions and vector-valued functions. First, we see that the derivative of a sum of two vectors is equal to the sum of their derivatives. The product rule also applies for scalar multiplication of a vector, for the dot product of a vector, and for the cross product of a vector. Pay special attention to interpret the quantities involved, though, as scalars or vectors. For example, recall the dot product gives us a scalar, while the cross product gives us a vector. Lastly, we see that the chain rule for the composition of a vector-valued function with a scalar function. Let's move on to consider tangent lines of vector-valued functions. One of the most important ideas in first semester Calculus is that a differentiable function is locally linear. That is, when viewed up close, the curve generated by a differentiable function looks very much like a line. In the same way, we expect that a smooth curve in three-space will look locally linear. Pictured here is a vector-valued function in red, and a tangent line to this curve in blue. Indeed, as we zoom in on a particular point, the curve begins to look indistinguishable from its tangent line. Just as we found tangent lines in single-variable Calculus, we can do the same for vector-valued functions. Recall that the vector equation of a line that passes through the point L0 in the direction of the vector u can be written as a function L of t that we have here. Using this and what we learned earlier about derivatives of vector-valued functions, we can then write the equation of the tangent line to the vector-valued function r at the input A as follows.