Home > Calculus > Differentiating Logarithmic Functions with Base e

How do you find the derivative of exponential function #y = e^9 ln x#?

Answer 1

Emily Ammerman

Product rule!

The product rule states that, for a function #y=f(x)g(x)#,

#(dy)/(dx)=f'(x)g(x)+f(x)g'(x)#

So, derivating your function:

#(dy)/(dx)=e^9ln(x)+e^9*(1/x)#

Thus,

#(dy)/(dx)=e^9(ln(x)+(1/x))#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Evelyn Agard

To find the derivative of the exponential function ( y = e^{9\ln(x)} ), you can use the chain rule. The derivative is ( \frac{dy}{dx} = e^{9\ln(x)} \cdot \frac{d}{dx}(9\ln(x)) ). Using the chain rule and the derivative of ( \ln(x) ), the derivative simplifies to ( \frac{dy}{dx} = e^{9\ln(x)} \cdot \frac{9}{x} ).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.