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

4. Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Business Calculus

4. Derivatives of 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

Step 1: Identify the function to differentiate. The given function is g(x) = x^4 + 4x + 4^x. This function consists of three terms: a polynomial term (x^4), a linear term (4x), and an exponential term (4^x).

Step 2: Differentiate the first term, x^4, using the power rule. The power rule states that d/dx[x^n] = n*x^(n-1). Applying this, the derivative of x^4 is 4x^3.

Step 3: Differentiate the second term, 4x. The derivative of a linear term ax is simply the constant a. Therefore, the derivative of 4x is 4.

Step 4: Differentiate the third term, 4^x. For exponential functions of the form a^x, the derivative is a^x * ln(a). Here, a = 4, so the derivative of 4^x is 4^x * ln(4).

Step 5: Combine the results from Steps 2, 3, and 4. The derivative of g(x) is g'(x) = 4x^3 + 4 + 4^x * ln(4).

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