Question
In Exercises 11 through 14 , find the total derivative $d u / d t$ by two methods: (a) Use the chain rule; (b) make the substitutions for $x$ and $y$ or for $x, y$, and $z$ before differentiating.$$u=\ln x y+y^{2} ; x=e^{t} ; y=e^{-t}$$
Step 1
We have: \[ u = \ln(xy) + y^2 \] with the substitutions: \[ x = e^t \quad \text{and} \quad y = e^{-t} \] Show more…
Show all steps
Your feedback will help us improve your experience
Subhadeepta Sahoo and 82 other Calculus 3 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
In Exercises 11-16, obtain the derivative $d y / d x$ and state the rules that you use. HINT [See Example 2.] $$ y=10 $$
Techniques of Differentiation with Applications
Derivatives of Powers, Sums, and Constant Multiples
In Exercises $13-16$ , calculate the derivative in two ways. First use the Product or Quotient Rule, then rewrite the function algebraically and apply the Power Rule directly. \begin{equation} f(t)=(2 t+1)\left(t^{2}-2\right) \end{equation}
DIFFERENTIATION
Product and Quotient Rules
In Exercises $13-24,$ find the derivative of $y$ with respect to the appropriate variable. $$ \begin{array}{l}{y=\left(x^{2}+1\right) \operatorname{sech}(\ln x)} \\ {\text { (Hint: Before differentiating, express in terms of exponentials and simplify.) }}\end{array} $$
Transcendental Functions
Hyperbolic Functions
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD