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

Question

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

y = 10^x(In 10^x-1)

Accepted Answer

What is the derivative of Y equals 12X multiplied by natural log of 12X minus 1? Where we have 4 possible answers, and all of these are functions with an exponential function, multiplying a log. Now, to solve this, we first need to use the product rule. The product rule of derivatives. Tells us if we have a function u multiplying V. Its derivative is going to be U multiplied by V. Plus you multiplied by V. Now, F of X is some function, you multiplying V. And in our case our u is our first equation. 12 X. V, in our case, is the other function, natural log, 12 is the X minus 1. Now we just need U and V. Now, U, in our case, goes back to U, and U is an exponential function with a base B. So, the derivative of a base exponential B to X is given by B to X, multiplied by natural log of B. This gives us 12 Rass multiplied. By natural log of 12. Or, we actually have a chain rule. Our outside function is natural luck. Now the derivative of natural luck. Is 1 divided by x. So we will write this as 1 divided by 12 ways to the X. We now multiply this by the derivative of the inside function, which, as we mentioned before, is 12 Ra CDX, natural log of 12. The derivative of the 1 just ends up being zero, because it's a derivative of a constant. We can now simplify. For this 12 to the X cancels out V1 divided by 12 X. Meaning we just lift with natural log of 12. We can then compose our product rule. F of X is equals to u. 12 CDX, natural log 12. Multiplying V Natural luck 12 to the X -1 Plus you. 12 to the X, multiplying V natural lag. Of 12. So, that's multiplying theme. Now we can simplify. We get 12 to the X natural log. 12 Multiplied By natural luck 12 reds to the X. Minus 12 to the X. Naturalo 12. Multiplying what? Which just gives us itself, 12 to the X, natural log 12. We will then add 12 to the X, natural log, 12. We can then see -12 X, natural 12, and 12 X 12 cancel each other out. We now have 12 to the X. Natural log 12 multiplied by natural log, 12 race to the X. Now we can make use of one of our log properties, which tells us if we have A lug Of some value raised to an exponent. Can rewrite it As the exponent multiplying natural log of that value. Which means we get 12 to the X, natural log 12 multiplied by X, natural log 12. Finally, we can combine our natural log 12s to get 12 to the X. Multiplied by X, multiplying natural log squared. Of 12 We can then see the answer to our problem. Corresponds with answer D. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

15–48. Derivatives Find the derivative of the following functions.y = 10^x(In 10^x-1)

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15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)