Find the derivative of the given function.
Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 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
- 10. Physics Applications of Integrals 3h 16m
- 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
- 16. Parametric Equations & Polar Coordinates7h 58m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
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Join thousands of students who trust us to help them ace their exams!Watch the first videoMultiple Choice
Identify the local minimum and maximum values of the given function, if any.
f(t)=t2lnt, t>0
A
Local maximum of 1 at x=1, Local minimum of −0.19 at x=21
B
Local maximum of −2e1 at x=e1 , No local minima
C
Local minimum of −2e1 at x=e1 , No local maxima
D
No local extrema

1
To find the local extrema of the function \( f(t) = t^2 \ln t \), we first need to find its derivative. Use the product rule for differentiation, which states that if \( u(t) = t^2 \) and \( v(t) = \ln t \), then \( (uv)' = u'v + uv' \).
Calculate the derivatives: \( u'(t) = 2t \) and \( v'(t) = \frac{1}{t} \). Substitute these into the product rule to find \( f'(t) = 2t \ln t + t^2 \cdot \frac{1}{t} = 2t \ln t + t \).
Set the derivative \( f'(t) = 2t \ln t + t \) equal to zero to find critical points: \( 2t \ln t + t = 0 \). Factor out \( t \) to get \( t(2 \ln t + 1) = 0 \). Since \( t > 0 \), we solve \( 2 \ln t + 1 = 0 \).
Solve \( 2 \ln t + 1 = 0 \) to find \( \ln t = -\frac{1}{2} \). Exponentiate both sides to solve for \( t \): \( t = e^{-\frac{1}{2}} = \frac{1}{\sqrt{e}} \). This is the critical point.
To determine if this critical point is a local minimum or maximum, use the second derivative test. Find \( f''(t) \) and evaluate it at \( t = \frac{1}{\sqrt{e}} \). If \( f''(t) > 0 \), it's a local minimum; if \( f''(t) < 0 \), it's a local maximum. Calculate \( f''(t) \) and substitute \( t = \frac{1}{\sqrt{e}} \) to conclude.
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Multiple Choice
Derivatives of Exponential & Logarithmic Functions practice set
