7–28. Derivatives Evaluate the following derivatives.

Question

d/dx ((2x)⁴ˣ)

Accepted Answer

Welcome back every once to another video. Differentiate the following function with respect to X. F of X is equal to 3 X2 all the way to the power of 2 X. So for this problem, we have to notice that our function has a base that is a function of X, and our exponent is also a function of X. So whenever this is the case, we're going to use logarithmic differentiation. And the first step is to assigny to our function. So Y equals 3x2 all raises the power of 2 X. And we're going to take the natural logarithm of both sides so that LN of Y is equal to LN of 3x2 raises the power of 2 X. Applying the properties of logarithms, we can show that LN of Y equals 2 X. LN of 3 X squared, right? So we're applying the power rule. And now we are ready to differentiate both sides of this relationship. So on the left hand side, a lot of Y is going to be 1 divided by Y multiplied by Y our derivative based on the chain rule, and on the right hand side, we want to differentiate a product. So we can factor out 2, and then our derivative would be X multiplied by LN of 3 X squared. Now, let's apply the product rule. So, first of all, we're going to rewrite the left-hand side, that's Y divided by Y. On the right hand side, we have 2. And then we're multiplying by the derivative of X, which is 1 multiplied by the function LN of 3x2. So that's LN of 3 X2 plus X. Multiplied by the derivative of LN of 3 x 2, so we have to apply the chain rule. First of all, we're taking 1 divided by 3 X2, and then we're multiplying by the derivative of 3 X2, right? According to the chain rule, that would be 6 X. Well done. So now, our goal is to simplify the left-hand side. Let's focus on the left hand side first. So, we have 2 multiplied by LN of 3 X 2 plus. Now, notice that X multiplied by X is X2, and X2 divided by X2, while those can be canceled out, and 6 divided by 3 is simply 2. So we're adding 2, right? So we have our expression on the right hand side, and we can now express Y, which is our derivative. We're going to multiply both sides by Y. we get Y equals 2 Y multiplied by LN of 3 X2 + 2. Now let's substitute for why we got 2 multiplied by. 3 X squared all raise the power of 2 X. And our final factor is LN of 3 X2 + 2. From here, we can simplify it slightly. So Y is equal to 2 multiplied by. Now, 3 raises the power of 2 X is going to be equal to 9 to the power of X, right? Because we can square 3 and basically, the remaining exponent is going to be X. And then we can raise X2 to the power of 2 X. Which would be X to the power of 4 X, right? 2 multiplied by 2 X is 4 X. And finally, our remaining factor is Alan of 3 X2 + 2. That'd be our final answer and thank you for watching.

7–28. Derivatives Evaluate the following derivatives.

d/dx ((2x)⁴ˣ)

Key Concepts

Derivatives

Product Rule

Exponential Functions

Watch next