15–48. Derivatives Find the derivative of the following functi...

Question

15–48. Derivatives Find the derivative of the following functions.

y = 10^In 2x

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. For the given function, find the derivative. Y equals 10 to the power of the natural log of 71 X. Awesome. So it appears for this particular problem we're asked to figure out what the derivative of this function of Y is. So now that we know that we're ultimately trying to figure out what the derivative of this function of Y is, our first step that we need to take is we need to note once again that we're trying to figure out the derivative of the given function, which our function once again is Y equals 10 to the power of the natural log of 71 X. And in order to help us solve for the derivative, we must first simplify the expression by recalling and using our properties of logarithms and exponents. And when we do that, we can take our function and we could simplify it as follows. We could state that Y. is equal to 10 to the power of the natural log of 71 X. Awesome, and this could be equal to. E to the power of the natural log of And Multiplied by the natural log of 71 X. Fantastic. So now, we need to note that sense, and let's make a note of this. That the natural log of 10 is equal to a constant value or C denotes constant, we can rewrite our expression as Y equals 71 X. To the power of, so we could say Y equals 71 X to the power of the natural log of 10. So now, our next step that we need to take is we need to recall and apply the chain rule and the power rule. So when we now differentiate Y equals 71 X to the power of LN 10 or the natural log of 10, we will need to use the power rule of differentiation for functions in the form of, and let's make a note of this, U to the power of N. So we're now using the form U to the power of N. Awesome. So with that in mind, we now need to go ahead and write that. D by DX since we're taking the derivative of X, so we're we're taking the derivative with our emphasis on X, so we're finding X. So D by DX of U to the power of N is equal to N multiplied by U to the power of N minus 1 multiplied by DU by DX. Fantastic. And let us note for this particular problem that we're solving for, we're gonna let you equal 71 X and we're gonna let N equal the natural log of 10. Fantastic. So now we could differentiate as follows. So we can go ahead and write that. DY by DX. is equal to the natural log. Of 10, multiplied by 71 X to the power of the natural log of 10 minus 1. Multiplied by D by DX of 71 X, and we could differentiate 71 X and the derivative of 71 X with respect to X will be 71, which we could write as D by DX of 71 X is equal to 71, and putting everything together, we could go ahead and write that D Y. So once again, we're combining all of our known answers into our final expression for the derivative. So we could write that DY by DX is equal to the natural log of 10, multiplied by 71. Multiplied by 71 X to the power of the natural log of 10 minus 1, and that's it. That is our final answer. Hooray, we did it. So therefore, without a doubt, our final answer for the derivative of this function of Y will be Dy by DX is equal to. And rather we could simplify, let's write this. So our final answer is DY by DX is equal to 71, multiplied by the natural log of 10, multiplied by 71 X to the power of the natural log of 10 minus 1. So you could write it either way. So, we could write it as we did above, or we could write it as so, so D Y equals DX. So D by DX is equal to 71 multiplied by the natural log of 10 multiplied by 71 X to the power of natural log of 10 minus 1, and either of those are our final answer. Final answers. Hooray, we did it. So this problem is fairly easy to solve for as long as you have a strong understanding and you're able to recall and use the proper differentiation tactics to solve this problem. So you need to make sure you know the chain rule, and you need to make sure you know how to take basic derivatives. And you also very, it's very important to understand the properties of logarithms and exponents, and you also need to be able to recall and use the power rule. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

15–48. Derivatives Find the derivative of the following functions.y = 10^In 2x

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15–48. Derivatives Find the derivative of the following functions.
y = 10^In 2x