Find the derivative of the given function.g(x)=x4+4x+4xg(x)=x^4+4x+4^xg(x)=x4+4x+4x

4 x 3 + 4 + 4 x ln ⁡ ⁡ 4 4x^3+4+4^{x}\ln⁡4 4 x 3 + 4 + 4 x ln ⁡4

Find the derivative of the given function. g ( x ) = x 4 + 4 x + 4 x g(x)=x^4+4x+4^x g ( x ) = x 4 + 4 x + 4 x

Here, we wanna find the derivative of the function g of x is equal to x to the power of 4 plus 4x plus 4 to the power of x. Now all of these terms have a 4 and they have an x, but their derivatives are not all the same because these are all types of functions. Right? Now since these are all being added together, that means that we can look at their individual derivatives and just add all of those together. So let's go ahead and get started in finding our derivative here, g prime of x. Now looking at that first term, x to the power of 4, here, I need to use the power rule. So I'm pulling that 4 out front, multiplying by it to give me 4 x, and then decreasing that power by 1. So that is 4 x cubed. Then for this next term, I'm adding its derivative together plus 4x or the derivative of 4x, which is just 4. Then finally, I have this 4 to the power of x. This is a general exponential function of the form b to the power of x. We know that the derivative of b to the x is going to be b to the x times the natural log of that base b. So since my base here is 4, this is going to be 4 to the x times the natural log of 4. And this is my full derivative here. Now we can see that all of these different functions have exactly the same elements. They have a 4, and they have an x. But taking their derivatives is wildly different. Let me know if you have any questions here, and I'll see you in the next one.

Find the derivative of the given function. g ( x ) = x 4 + 4 x + 4 x g(x)=x^4+4x+4^x g ( x ) = x 4 + 4 x + 4 x

4 x 3 + 4 + 4 x ln ⁡ ⁡ 4 4x^3+4+4^{x}\ln⁡4 4 x 3 + 4 + 4 x ln ⁡4

4 x 3 + 4 x x − 1 4x^3+4x^{x-1} 4 x 3 + 4 x x − 1

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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Multiple Choice

Find the derivative of the given function.
$g(x)=x^4+4x+4^x$

$12x$

$4x^3+8$

$4x^3+4x^{x-1}$

$4x^3+4+4^{x}\ln4$

Verified step by step guidance

Identify the function for which you need to find the derivative: $ g(x) = x^4 + 4x + 4^x $.

Apply the power rule to differentiate $ x^4 $. The power rule states that $ \frac{d}{dx}[x^n] = nx^{n-1} $. So, the derivative of $ x^4 $ is $ 4x^3 $.

Differentiate the linear term $ 4x $. The derivative of $ 4x $ is simply $ 4 $, as the derivative of $ x $ is 1.

Differentiate the exponential term $ 4^x $. Use the formula for the derivative of an exponential function $ a^x $, which is $ a^x \ln(a) $. Therefore, the derivative of $ 4^x $ is $ 4^x \ln(4) $.

Combine all the derivatives obtained from each term to get the final derivative: $ g'(x) = 4x^3 + 4 + 4^x \ln(4) $.

4:50m

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