Question Video: Differentiating the Sum of Two Exponential and Logarithmic Functions Using the Chain Rule Mathematics

Video Transcript

Determine the first derivative of 𝑦 is equal to negative 𝑒 to the power of 𝑥 over four plus two times the natural logarithm of 𝑥 to the eighth power.

We’re given that 𝑦 is the sum of an exponential function and the natural logarithm of a polynomial. And we’re asked to calculate the first derivative of this function. Since 𝑦 is a function of 𝑥, this is asking us to calculate the derivative of 𝑦 with respect to 𝑥. So we need to calculate the derivative of negative 𝑒 to the power of 𝑥 over four plus two times the natural logarithm of 𝑥 to the eighth power.

To start, if we’re asked to differentiate the sum of two functions, we can differentiate them separately and then add the results. This gives us our derivative is equal to the derivative of negative 𝑒 to the power of 𝑥 over four. With respect to 𝑥 plus the derivative of two times the natural logarithm of 𝑥 to the eighth power with respect to 𝑥.

We’re now ready to differentiate our first term by using the fact that, for any constants 𝑎 and 𝑛, the derivative of 𝑎𝑒 to the power of 𝑛𝑥 with respect to 𝑥 is equal to 𝑎𝑛𝑒 to the power of 𝑛𝑥. We multiplied by the coefficient in our exponent. So the derivative of our first term is negative a quarter 𝑒 to the power of 𝑥 over four.

To differentiate our second term, we might be tempted to use the chain rule. Since we see that our second term is the composition of a natural logarithm function and a polynomial. However, there’s a simpler way by using one of our laws for logarithms. The power law for logarithms tells us that 𝑎 times the logarithm of 𝑥 to the 𝑛th power is equal to 𝑎 times 𝑛 times the logarithm of 𝑥.

Since the natural logarithm function is a logarithmic function, we can use the power law for logarithms to change this into two times eight times the natural logarithm of 𝑥. We have that two multiplied by 𝑎 is equal to 16. And since this is a constant with respect to 𝑥, we can take the constant factor of 16 outside of our derivative.

So the last thing we have to do is evaluate the derivative of the natural logarithm of 𝑥 with respect to 𝑥. And we know the derivative of the natural logarithm of 𝑥 with respect to 𝑥 is just equal to one divided by 𝑥. Therefore, we’ve shown the first derivative of 𝑦 is equal to negative 𝑒 to the power of 𝑥 over four. Plus two times the natural logarithm of 𝑥 to the eighth power is equal to negative a quarter 𝑒 to the power of 𝑥 over four plus 16 divided by 𝑥.