Find the indicated derivative.h(t)=12t3+4t2+3th\left(t\right)=\frac12t^3+\frac{4}{t^2}+3\sqrt{t}h′(t)=h^{\prime}(t)=

32t2−8t3+32t\$$frac32t^2$$-\frac{8}{$$t^3$$}+\frac{3}{2\sqrt{t}}

Find the indicated derivative. h ( t ) = 1 2 t 3 + 4 t 2 + 3 t h\left(t\right)=\frac12t^3+\frac{4}{t^2}+3\sqrt{t} h ′ ( t ) = h^{\prime}(t)=

In this problem, we're asked to find the derivative of the function h of t equals 1 half t cubed plus 4 over t squared plus 3 root t. So let's go ahead and get started in finding this derivative. Now looking at this function, I know that I wanna go ahead and rewrite a couple of these terms as power functions. So I have this one half t cubed. That is fine as is one half t cubed. It's really easy to see how we would use the power rule here. But for that second term, since I'm dealing with a fraction with my variable on the bottom, in order to make that variable on the top, I can just make that power negative. So the second term is 4 t to the power of negative 2, and that's gonna make it easier to apply the power rule. Now for my last term, I have 3 root t. Remember that the square root of anything is really just to the power of 1 half. So I can rewrite this as 3 times t to the power of 1 half, which just means I'm taking the square root there. Now it's much more easy to see for both of these terms that we can use the power rule. So let's go ahead and apply that power rule in finding the derivative. So let's start with that first term. I have 1 half t cubed. Now here, of course, wanna use the power rule, which means I'm gonna pull that 3 out to the front and decrease that exponent by 1. Now remember, because we already have this constant 1 half in front, I need to make sure that I'm accounting for that when I'm using the power rule. So when I pull that 3 out to the front, I wanna think of it as multiplying my original constant 1 half. So 3 times 1 half is going to give me 3 halves. So I have 3 halves times t to the power of 3 minus 1, which is just 2. Now moving on to my next term here, I have 4 t to the power of negative 2. Again, here, we wanna use the power rule. So I'm gonna pull that negative 2 out to the front and decrease that exponent by 1. Again, since I already have this 4 out front, I know that I'm taking that 4. I'm multiplying it by that negative 2 that I'm pulling out using the power rule. So that makes this a negative eight times t to the power of negative two minus one, which is negative 3. Now for my last term here, I have 3 times t to the power of 1 half. Again, using the power rule here, I'm gonna pull that one half to the front, decrease that power by 1, 3 times 1 half taking into account my original constant. And that one half I'm pulling out to the front gives me positive 3 halves times t to the power of 1 half minus 1, which is negative one half. Now this answer is already correct as is, but I'm gonna go ahead and rewrite this to more closely match how our original function was given to us. And this is typically what you're going to wanna do when presenting your final answers. Since our original function had 4 over t squared as opposed to 4 t to the power of negative 2, I'm going to go ahead and rewrite any of my negative powers back as a fraction. So let's go ahead and do that here. Now that first term is going to stay the same, three halves t squared. That's fine as is. And this is minus 8 over t cubed, taking away that negative, putting it back on the bottom of our fraction. Then for this next term, I have 3 over 2 and this is t to the power of negative one half. Now t to the power of 1 half we know is the square root of t. When it's a negative that just means it's on the bottom of the fraction. So I have t to the power of 1 half on the bottom which I can just rewrite as the square root of t on the bottom here. So this is my final answer in its final form. 3 has t squared minus 8 over t cubed plus 3 over 2 root t, and that's the derivative of my original function. Let me know if you have any questions, and I'll see you in the next one.

Find the indicated derivative.h(t)=12t3+4t2+3th\left(t\right)=\$$frac12t^3$$+\frac{4}{$$t^2$$}+3\sqrt{t}h′(t)=$$h^{\prime}(t)=$$

32t2−8t3+32t\$$frac32t^2$$-\frac{8}{$$t^3$$}+\frac{3}{2\sqrt{t}}

t2−$$2t3+1tt^2$$-\frac{2}{$$t^3$$}+\frac{1}{\sqrt{t}}t2−t32+t1

32t2−8t3+32t\$$frac32t^2$-8t^3$$+\frac32\sqrt{t}23t2−8t3+23t

32t3−8t2+32t\$$frac32t^3$$-\frac{8}{$$t^2$$}+\frac32\sqrt{t}

Table of contents

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0. Functions7h 55m

Introduction to Functions
18m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 5m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

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 31m

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
52m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

13. Intro to Differential Equations2h 55m

Basics of Differential Equations
48m

Slope Fields
19m

Euler's Method
22m

Separable Differential Equations
1h 25m

14. Sequences & Series5h 36m

Sequences
1h 22m

Review of Factorials
11m

Series
44m

Convergence Tests
3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

3. Techniques of Differentiation

Basic Rules of Differentiation

Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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Multiple Choice

Find the indicated derivative.
$h (t) = \frac{1}{2} t^{3} + \frac{4}{t^{2}} + 3 \sqrt{t}$
$h^{'} (t) =$

$t^2-$\frac{2}{t^3}$+$\frac{1}{\sqrt{t}$}$

$\frac\)32t^2-8t^3+$\frac$32\(\sqrt{t}$

$\frac{3}{2} t^{2} - \frac{8}{t^{3}} + \frac{3}{2 \sqrt{t}}$

$\frac{3}{2} t^{3} - \frac{8}{t^{2}} + \frac{3}{2} \sqrt{t}$

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Verified step by step guidance

Identify the function h(t) that needs to be differentiated. The function is given as h(t) = $\frac{1}{2}$t^3 + $\frac{4}{t^2}$ + 3$\sqrt{t}$.

Rewrite the function in a form that is easier to differentiate. This involves expressing each term with exponents: h(t) = $\frac{1}{2}$t^3 + 4t^{-2} + 3t^{1/2}.

Apply the power rule to differentiate each term separately. The power rule states that if f(t) = t^n, then f'(t) = nt^{n-1}.

Differentiate the first term: $\frac{1}{2}$t^3 becomes $\frac{3}{2}$t^2.

Differentiate the second term: 4t^{-2} becomes -8t^{-3}, and the third term: 3t^{1/2} becomes $\frac{3}{2}$t^{-1/2}.

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Struggling with Calculus?

Find the indicated derivative.h(t)=12t3+4t2+3th\(\left\)(t\(\right\))=\(\frac\)12t^3+\(\frac{4}{t^2}\)+3\(\sqrt{t}\)h′(t)=h^{\(\prime\)}(t)=

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Basic Rules of Differentiation practice set

Find the indicated derivative.
$h (t) = \frac{1}{2} t^{3} + \frac{4}{t^{2}} + 3 \sqrt{t}$
$h^{'} (t) =$