Find the derivative of the given function.g(x)=ex2lnxg(x)=e^{x^2\ln x}g(x)=ex2lnx

xex2ln⁡x(2ln⁡$$x+1)xe^{x^2\ln x}$$\left(2\ln x+1\right)xex2lnx(2lnx+1)

Find the derivative of the given function. g ( x ) = e x 2 ln x g(x)=e^{x^2\ln x} g ( x ) = e x 2 l n x

Here, we wanna find the derivative of the function g of x is equal to e to the power of x squared times the natural log of x. Now here, we're going to have to utilize the chain rule. Most of you should already know the chain rule, so we can continue as normal. But if you have not learned the chain rule yet, you're gonna wanna circle back to this problem once you've learned it. For the rest of you, let's get into finding our derivative, g prime of x. So starting with our outermost function, this is e to the power of this x squared times the natural log of x. We know that the derivative of our basic exponential function e to the x is just going to be e to the x. But here, since that power is not just x, it is x squared times the natural log of x, I need to put all of that still in my power. Now multiplying by the derivative of this x squared natural log of x, that's what our chain rule tells us to do. Because here, I have a function multiplying another function, x squared times the natural log of x, I need to apply the product rule. So we're not only using the chain rule. We're also using the product rule within this chain rule. So multiplying here by the derivative of x squared natural log of x. Remember our product rule memory tool here, left d right plus right d left, so that left hand function x squared times the derivative of that right hand function, which is one over x, then plus right d left, that is the natural log of x, times the derivative of x squared, that is just 2 x. Now the all of our calculus is done here. We've taken our derivative. We need to simplify this a little bit because there are some things that will cancel. Now we have this x and this x squared, and then we have 2x times the natural log of x. Let's write this a little bit neater. That is e to the x squared times the natural log of x times x plus 2 X natural log of X. Now both of these terms have an X in them, so we can go ahead and factor that out as well just to fully simplify this. This is X times e to the X squared times the natural log of X times 1 plus 2 natural log of x. And this is our final derivative here, having applied all of the rules that we know for derivatives. Let us know if you have any questions, and let's keep going.

Find the derivative of the given function.g(x)=ex2ln⁡xg(x)=$$e^{x^2\ln x}g(x)=ex2$$lnx

xex2ln⁡x(2ln⁡$$x+1)xe^{x^2\ln x}$$\left(2\ln x+1\right)xex2lnx(2lnx+1)

ex2ln⁡$$xe^{x^2\ln x}ex2$$lnx

$$e2x+1xe^{2x+\frac{1}$${x}}e2x+x1

2ex2ln⁡$$x2e^{x^2\ln x}2ex2$$lnx

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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

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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

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1h 13m

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Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
52m

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2h 32m

Partial Fractions
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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객관식

Find the derivative of the given function.
$g(x)=e^{x^2$\ln$ x}$

$e^{x^2$\ln$ x}$

$e^{2x+$\frac{1}{x}$}$

$xe^{x^2$\ln$ x}$\left$(2$\ln$ x+1$\right$)$

$2e^{x^2$\ln$ x}$

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Identify the function for which you need to find the derivative: $ g(x) = e^{x^2 \ln x} $. This is an exponential function where the exponent is a function of $ x $.

Use the chain rule to differentiate $ g(x) = e^{u(x)} $, where $ u(x) = x^2 \ln x $. The chain rule states that $ \frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot \frac{du}{dx} $.

Find $ \frac{du}{dx} $ for $ u(x) = x^2 \ln x $. Use the product rule, which states that $ \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) $. Here, $ f(x) = x^2 $ and $ g(x) = \ln x $.

Calculate $ f'(x) = \frac{d}{dx}(x^2) = 2x $ and $ g'(x) = \frac{d}{dx}(\ln x) = \frac{1}{x} $. Apply the product rule: $ \frac{du}{dx} = 2x \ln x + x^2 \cdot \frac{1}{x} = 2x \ln x + x $.

Substitute $ \frac{du}{dx} $ back into the chain rule expression: $ \frac{d}{dx} g(x) = e^{x^2 \ln x} \cdot (2x \ln x + x) $. This gives the derivative of the function.