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

Determine the first derivative of π¦ = βπ^(π₯/4) + 2 ln π₯βΈ.

02:20

### Video Transcript

Determine the first derivative of π¦ is equal to negative π to the power of π₯ over four plus two times the natural logarithm of π₯ to the eighth power.

Weβre given that π¦ is the sum of an exponential function and the natural logarithm of a polynomial. And weβre asked to calculate the first derivative of this function. Since π¦ is a function of π₯, this is asking us to calculate the derivative of π¦ with respect to π₯. So we need to calculate the derivative of negative π to the power of π₯ over four plus two times the natural logarithm of π₯ to the eighth power.

To start, if weβ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 π to the power of π₯ over four. With respect to π₯ plus the derivative of two times the natural logarithm of π₯ to the eighth power with respect to π₯.

Weβre now ready to differentiate our first term by using the fact that, for any constants π and π, the derivative of ππ to the power of ππ₯ with respect to π₯ is equal to πππ to the power of ππ₯. We multiplied by the coefficient in our exponent. So the derivative of our first term is negative a quarter π to the power of π₯ 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, thereβs a simpler way by using one of our laws for logarithms. The power law for logarithms tells us that π times the logarithm of π₯ to the πth power is equal to π times π times the logarithm of π₯.

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 π₯. We have that two multiplied by π is equal to 16. And since this is a constant with respect to π₯, 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 π₯ with respect to π₯. And we know the derivative of the natural logarithm of π₯ with respect to π₯ is just equal to one divided by π₯. Therefore, weβve shown the first derivative of π¦ is equal to negative π to the power of π₯ over four. Plus two times the natural logarithm of π₯ to the eighth power is equal to negative a quarter π to the power of π₯ over four plus 16 divided by π₯.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.