Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

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3:15 minutes

Problem 7.R.13

Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(t) = cosh t sinh t

Verified step by step guidance

Recall the product rule for derivatives: if you have a function $f(t) = u(t) v(t)$, then its derivative is $f'(t) = u'(t) v(t) + u(t) v'(t)$.

Identify the two functions in the product: $u(t) = \cosh t$ and $v(t) = \sinh t$.

Find the derivatives of each function separately: $\frac{d}{dt} \cosh t = \sinh t$ and $\frac{d}{dt} \sinh t = \cosh t$.

Apply the product rule: $f'(t) = (\sinh t)(\sinh t) + (\cosh t)(\cosh t)$.

Simplify the expression if possible, using hyperbolic identities such as $\cosh^2 t - \sinh^2 t = 1$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Functions

Hyperbolic functions, such as sinh(t) and cosh(t), are analogs of trigonometric functions but based on hyperbolas. They have unique properties and identities, like cosh²(t) - sinh²(t) = 1, which are useful in differentiation and integration.

Product Rule for Differentiation

The product rule states that the derivative of a product of two functions u(t) and v(t) is u'(t)v(t) + u(t)v'(t). This rule is essential when differentiating functions like f(t) = cosh(t) * sinh(t), where both factors depend on t.

Derivatives of Hyperbolic Functions

The derivatives of basic hyperbolic functions are: d/dt[sinh(t)] = cosh(t) and d/dt[cosh(t)] = sinh(t). Knowing these derivatives allows you to apply the product rule correctly to find the derivative of functions involving hyperbolic terms.