sepment \( \mathrm{OCl} \) (Cikk en \( \frac{3}{2} t \). and choos \( f f d x \) we ener ene inograt)
Added by Baddest P.
Close
Step 1
What is it that we need to solve or understand? Once we have a clear understanding of the problem, let's gather all the relevant information or data that is available. This could involve conducting research, analyzing existing data, or seeking input from others Show more…
Show all steps
Your feedback will help us improve your experience
Eleni Katirtzoglou and 57 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let $f(x, y)=e^{x y}$ and $c(t)=\left(t^{3}, 1+t\right).$ \begin{equation}\begin{array}{l}{\text { (a) Calculate } \nabla f \text { and } c^{\prime}(t) \text { . }} \\ {\text { (b) Use the Chain Rule for Paths to calculate } \frac{d}{d t} f(c(t)) \text { . }} \\ {\text { (c) Write out the composite } f(c(t)) \text { as a function of } t \text { and differentiate. }} \\ {\text { Check that the result agrees with part (b). }}\end{array}\end{equation}
DIFFERENTIATION IN SEVERAL VARIABLES
The Gradient and Directional Derivatives
Let $f(x, y)=e^{x y}$ and $\mathbf{r}(t)=\left\langle t^{3}, 1+t\right\rangle$ $$\begin{array}{l}{\text { (a) Calculate } \nabla f \text { and } \mathbf{r}^{\prime}(t) .} \\ {\text { (b) Use the Chain Rulc for Paths to calculate } \frac{d}{d t} f(\mathbf{r}(t)) \text { . }} \\ {\text { (c) Write out the composite } f(\mathbf{r}(t)) \text { as a function of } t \text { and differentiate. }} \\ {\text { Check that the result agrees with part (b). }}\end{array}$$
$\mathbf{F}(t)=\mathbf{f}(u(t)) .$ Find $\mathbf{F}^{\prime}(t)$ in terms of $t.$ $$\mathbf{f}(u)=\cos u \mathbf{i}+e^{3 u} \mathbf{j} \text { and } u(t)=3 t^{2}-4$$
Geometry in Space and Vectors
Vector-Valued Functions and Curvilinear Motion
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Watch the video solution with this free unlock.
EMAIL
PASSWORD