Question Video: Differentiating a Combination of Logarithm Arithmic and Polynomial Functions Using the Quotient Rule | Nagwa Question Video: Differentiating a Combination of Logarithm Arithmic and Polynomial Functions Using the Quotient Rule | Nagwa

Question Video: Differentiating a Combination of Logarithm Arithmic and Polynomial Functions Using the Quotient Rule Mathematics • Third Year of Secondary School

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Find d𝑦/dπ‘₯, given that 𝑦 = 9π‘₯/ln 9π‘₯.

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Video Transcript

Find d𝑦 by dπ‘₯ given that 𝑦 is equal to nine π‘₯ divided by the natural logarithm of nine π‘₯.

The question wants us to find d𝑦 by dπ‘₯. That’s the first derivative of 𝑦 with respect to π‘₯. And we can see that 𝑦 is the quotient of two functions. It’s the quotient of nine π‘₯ and the natural logarithm of nine π‘₯. So we’ll find this derivative by using the quotient rule. We recall the quotient rule tells us if 𝑦 is the quotient of two functions 𝑒 over 𝑣, then d𝑦 by dπ‘₯ is equal to 𝑣 times d𝑒 by dπ‘₯ minus 𝑒 times d𝑣 by dπ‘₯ all divided by 𝑣 squared.

So to use the quotient rule, we’ll start by setting 𝑒 of π‘₯ to be the function in our numerator, that’s nine π‘₯, and 𝑣 of π‘₯ to be the function in our denominator, that’s the natural logarithm of nine π‘₯. And to apply the quotient rule, we’re going to need to find expressions for d𝑒 by dπ‘₯ and d𝑣 by dπ‘₯. Let’s start with finding d𝑒 by dπ‘₯. That’s the derivative of nine π‘₯ with respect to π‘₯. And nine π‘₯ is just a linear function. So its derivative is the coefficient of π‘₯, which, in this case, is nine.

Let’s now find an expression for d𝑣 by dπ‘₯. That’s the derivative of the natural logarithm of nine π‘₯ with respect to π‘₯. And we can do this by using one of our standard derivative results for logarithmic functions. For any positive constant π‘Ž, the derivative of the natural logarithm of π‘Žπ‘₯ with respect to π‘₯ is equal to one divided by π‘₯. So in our case, the derivative of the natural logarithm of nine π‘₯ with respect to π‘₯ is equal to one divided by π‘₯. So we’re now ready to find the d𝑦 by dπ‘₯ by using the quotient rule.

The quotient rule tells us d𝑦 by dπ‘₯ will be equal to 𝑣 times d𝑒 by dπ‘₯ minus 𝑒 times d𝑣 by dπ‘₯ divided by 𝑣 squared. Substituting in our expressions for 𝑒, 𝑣, d𝑒 by dπ‘₯, and d𝑣 by dπ‘₯, we get d𝑦 by dπ‘₯ is equal to the natural logarithm of nine π‘₯ multiplied by nine minus nine π‘₯ times one over π‘₯ all divided by the natural logarithm of nine π‘₯ squared. And we can simplify this expression. First, we’ll cancel π‘₯ multiplied by one over π‘₯.

Next, we want to take out a factor of nine in our numerator. And this gives us nine times the natural logarithm of nine π‘₯ minus one all divided by the natural logarithm of nine π‘₯ squared. And this is our final answer. Therefore, we’ve shown if 𝑦 is equal to nine π‘₯ divided by the natural logarithm of nine π‘₯, then d𝑦 by dπ‘₯ is equal to nine times the natural logarithm of nine π‘₯ minus one all divided by the natural logarithm of nine π‘₯ squared.

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