Video Transcript
Determine the second derivative of 𝑦 with respect to 𝑥 given that 𝑥 is equal to six times the natural logarithm of 𝑛 to the fifth power and 𝑦 is equal to negative eight times 𝑛 cubed.
In this question, we’re given 𝑥 and 𝑦 defined parametrically. And we need to use this to determine the second derivative of 𝑦 with respect to 𝑥. And since 𝑥 is a function in 𝑛 and 𝑦 is a function in 𝑛, we’re going to need to do this by using parametric differentiation. To do this, we’ll start by recalling how to find the second derivative of 𝑦 with respect to 𝑥, where 𝑦 and 𝑥 are given parametrically. We know that d two 𝑦 by d𝑥 squared is equal to the derivative with respect to 𝑛 of d𝑦 by d𝑥 divided by the derivative of 𝑥 with respect to 𝑛.
Therefore, to find this derivative, we need to differentiate d𝑦 by d𝑥 with respect to 𝑛 and then divide this by the derivative of 𝑥 with respect to 𝑛. However, there’s one small problem. We don’t know the derivative of 𝑦 with respect to 𝑥 since 𝑦 is a function in 𝑛 and 𝑥 is a function in 𝑛. So we’re going to need to find this by using the chain rule. And to make this easier, we’ll recall slight variation of the chain rule which says that d𝑦 by d𝑥 will be equal to the derivative of 𝑦 with respect to 𝑛 divided by the derivative of 𝑥 with respect to 𝑛.
We can now use all of this information to find the second derivative of 𝑦 with respect to 𝑥. Let’s start by finding the first derivative of 𝑦 with respect to 𝑥. To do this, we need to find d𝑦 by d𝑛 and d𝑥 by d𝑛. Let’s start by finding the derivative of 𝑦 with respect to 𝑛. That’s the derivative of negative eight 𝑛 cubed with respect to 𝑛. Since this is a polynomial, we can do this by using the power rule for differentiation. We multiply by the exponent of 𝑛 and then reduce this exponent by one. We get negative eight multiplied by three times 𝑛 squared, which is negative 24 𝑛 squared.
We now want to find an expression for d𝑥 by d𝑛. We need to differentiate six times the natural logarithm of 𝑛 to the fifth power with respect to 𝑛. And we might be tempted to do this by using the chain rule since this is the composition of two functions. However, we can simplify this by using the power rule for logarithms. We’re taking the natural logarithm of 𝑛 to the fifth power. And the power rule for logarithms tells us if we’re taking the logarithm of a power function, we can instead multiply it by the exponent. And we know six times five is equal to 30. Therefore, we can rewrite this as the derivative of 30 times the natural logarithm of 𝑛 with respect to 𝑛.
We can then directly evaluate this by recalling the derivative of the natural logarithm function is the reciprocal function. So the derivative of 30 times the natural logarithm of 𝑛 with respect to 𝑛 is 30 over 𝑛. We can now find d𝑦 by d𝑥 by taking the quotient of these two expressions. This gives us the derivative of 𝑦 with respect to 𝑥 is equal to negative 24𝑛 squared divided by 30 over 𝑛. And we can simplify this. First, we can multiply the numerator and denominator by 𝑛. This gives us negative 24𝑛 cubed divided by 30.
Next, we can note that negative 24 and 30 share a factor of six. It cancels to give negative four over five, so d𝑦 by d𝑥 is equal to negative four-fifths 𝑛 cubed. Now that we have an expression for d𝑦 by d𝑥 and d𝑥 by d𝑛, we can use our formula to find an expression for d two 𝑦 by d𝑥 squared. Substituting our expressions for d𝑦 by d𝑥 and d𝑥 by d𝑛 into this formula, we get d two 𝑦 by d𝑥 squared is equal to the derivative of negative four-fifths 𝑛 cubed with respect to 𝑛 divided by 30 over 𝑛.
Let’s start by simplifying our numerator by evaluating the derivative. Once again, this is the derivative of a polynomial. So we’ll do this by using the power rule for differentiation. We multiply by the exponent of 𝑛, which is three, and reduce this exponent by one. This gives us negative twelve-fifths 𝑛 squared. And we still need to divide this by 30 over 𝑛. We now need to simplify this. Once again, we’ll start by multiplying both the numerator and denominator by 𝑛. This gives us negative twelve-fifths 𝑛 cubed divided by 30. Next, dividing by 30 is the same as multiplying by one over 30. This gives us the following expression.
Finally, we can notice that 12 and 30 both share a factor of six. So we get negative two over five multiplied by one over five, which is negative two twenty-fifths. This then gives us our final answer. If 𝑥 is equal to six times the natural logarithm of 𝑛 to the fifth power and 𝑦 is equal to negative eight 𝑛 cubed, then the second derivative of 𝑦 with respect to 𝑥 is equal to negative two 𝑛 cubed divided by 25.
