Question Video: Differentiating a Combination of Logarithm Arithmic and Polynomial Functions Using the Quotient Rule Mathematics • Higher Education

Find d๐‘ฆ/d๐‘ฅ, given that ๐‘ฆ = 9๐‘ฅ/ln 9๐‘ฅ.

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Video Transcript

Find d๐‘ฆ by d๐‘ฅ given that ๐‘ฆ is equal to nine ๐‘ฅ divided by the natural logarithm of nine ๐‘ฅ.

The question wants us to find d๐‘ฆ by d๐‘ฅ. Thatโ€™s the first derivative of ๐‘ฆ with respect to ๐‘ฅ. And we can see that ๐‘ฆ is the quotient of two functions. Itโ€™s the quotient of nine ๐‘ฅ and the natural logarithm of nine ๐‘ฅ. So weโ€™ll find this derivative by using the quotient rule. We recall the quotient rule tells us if ๐‘ฆ is the quotient of two functions ๐‘ข over ๐‘ฃ, then d๐‘ฆ by d๐‘ฅ is equal to ๐‘ฃ times d๐‘ข by d๐‘ฅ minus ๐‘ข times d๐‘ฃ by d๐‘ฅ all divided by ๐‘ฃ squared.

So to use the quotient rule, weโ€™ll start by setting ๐‘ข of ๐‘ฅ to be the function in our numerator, thatโ€™s nine ๐‘ฅ, and ๐‘ฃ of ๐‘ฅ to be the function in our denominator, thatโ€™s the natural logarithm of nine ๐‘ฅ. And to apply the quotient rule, weโ€™re going to need to find expressions for d๐‘ข by d๐‘ฅ and d๐‘ฃ by d๐‘ฅ. Letโ€™s start with finding d๐‘ข by d๐‘ฅ. Thatโ€™s the derivative of nine ๐‘ฅ with respect to ๐‘ฅ. And nine ๐‘ฅ is just a linear function. So its derivative is the coefficient of ๐‘ฅ, which, in this case, is nine.

Letโ€™s now find an expression for d๐‘ฃ by d๐‘ฅ. Thatโ€™s the derivative of the natural logarithm of nine ๐‘ฅ with respect to ๐‘ฅ. And we can do this by using one of our standard derivative results for logarithmic functions. For any positive constant ๐‘Ž, the derivative of the natural logarithm of ๐‘Ž๐‘ฅ with respect to ๐‘ฅ is equal to one divided by ๐‘ฅ. So in our case, the derivative of the natural logarithm of nine ๐‘ฅ with respect to ๐‘ฅ is equal to one divided by ๐‘ฅ. So weโ€™re now ready to find the d๐‘ฆ by d๐‘ฅ by using the quotient rule.

The quotient rule tells us d๐‘ฆ by d๐‘ฅ will be equal to ๐‘ฃ times d๐‘ข by d๐‘ฅ minus ๐‘ข times d๐‘ฃ by d๐‘ฅ divided by ๐‘ฃ squared. Substituting in our expressions for ๐‘ข, ๐‘ฃ, d๐‘ข by d๐‘ฅ, and d๐‘ฃ by d๐‘ฅ, we get d๐‘ฆ by d๐‘ฅ is equal to the natural logarithm of nine ๐‘ฅ multiplied by nine minus nine ๐‘ฅ times one over ๐‘ฅ all divided by the natural logarithm of nine ๐‘ฅ squared. And we can simplify this expression. First, weโ€™ll cancel ๐‘ฅ multiplied by one over ๐‘ฅ.

Next, we want to take out a factor of nine in our numerator. And this gives us nine times the natural logarithm of nine ๐‘ฅ minus one all divided by the natural logarithm of nine ๐‘ฅ squared. And this is our final answer. Therefore, weโ€™ve shown if ๐‘ฆ is equal to nine ๐‘ฅ divided by the natural logarithm of nine ๐‘ฅ, then d๐‘ฆ by d๐‘ฅ is equal to nine times the natural logarithm of nine ๐‘ฅ minus one all divided by the natural logarithm of nine ๐‘ฅ squared.

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