Textbook Question
5–8. Calculate dy/dx using implicit differentiation.
sin y+2 = x
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.4.43
Derivatives Find and simplify the derivative of the following functions.
g(t) = t³+3t²+t / t³
Verified step by step guidance1
Step 1: Simplify the function g(t) = \(\frac{t^3 + 3t^2 + t}{t^3}\) by dividing each term in the numerator by t^3. This gives g(t) = 1 + \(\frac{3}{t}\) + \(\frac{1}{t^2}\).
Step 2: Rewrite the function in terms of powers of t: g(t) = 1 + 3t^{-1} + t^{-2}.
Step 3: Differentiate each term separately using the power rule. The power rule states that the derivative of t^n is n*t^{n-1}.
Step 4: Apply the power rule: The derivative of 1 is 0, the derivative of 3t^{-1} is -3t^{-2}, and the derivative of t^{-2} is -2t^{-3}.
Step 5: Combine the derivatives to find g'(t) = 0 - 3t^{-2} - 2t^{-3}. Simplify to get g'(t) = -\(\frac{3}{t^2}\) - \(\frac{2}{t^3}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
Recommended video:
05:44
Derivatives
Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function in the form f(t) = u(t)/v(t), the derivative is given by f'(t) = (u'v - uv')/v², where u and v are differentiable functions of t. This rule is essential when dealing with functions that are divided by another function.
Recommended video:
06:43The Quotient Rule
Simplification of Derivatives
After finding the derivative of a function, simplification is often necessary to express the result in its simplest form. This may involve factoring, reducing fractions, or combining like terms. Simplifying the derivative can make it easier to analyze the function's behavior, such as identifying critical points or determining concavity.
Recommended video:
05:44Derivatives
Related Practice
Textbook Question
Evaluate the derivative of the following functions.
f(y) = tan-1 (2y2 - 4)
Textbook Question
A spherical balloon is inflated and its volume increases at a rate of 15 in³/min. What is the rate of change of its radius when the radius is 10 in?
Textbook Question
Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/4 cot x−1 / x−π/4
Textbook Question
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = x^2/3, for x>0
Textbook Question
State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?
2
views