# Confused with when to use chain rule

Just some general questions as I'm confused with when to use chain rule when not to.

For instance, to find the derivative of e^sqrt(x), the right answer is to use chain rule to get e^sqrtx*the derivative of sqrt(x). BUT, isn't there a formula that: d/dx K^x = In(K)*K^x? K for constant and x for differentiable function. So why I can't use it to get e^sqrt(x)=Ine*e^sqrt(x)? AND, isn't d/dx(e^x) = e^x?? I'm completely confused.. Midterm tomorrow.. really need your help.. Thanks a lot!

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Mark44
Mentor
Just some general questions as I'm confused with when to use chain rule when not to.

For instance, to find the derivative of e^sqrt(x), the right answer is to use chain rule to get e^sqrtx*the derivative of sqrt(x). BUT, isn't there a formula that: d/dx K^x = In(K)*K^x?
No, there isn't. There is no function named In. Are you thinking ln (LN) for the natural logarithm?
K for constant and x for differentiable function. So why I can't use it to get e^sqrt(x)=Ine*e^sqrt(x)? AND, isn't d/dx(e^x) = e^x?? I'm completely confused.. Midterm tomorrow.. really need your help.. Thanks a lot!
d/dx(ex) = ex, but what about d/dx(ef(x))? For that you need the chain rule.

Let u = f(x).
d/dx(ef(x)) = d/dx(eu) = d/du(eu) * du/dx = eu * du/dx.

I don't know what you mean by In(K). Do you mean ln(K) (natural logarithm of K)? Assuming that's ln:

BUT, isn't there a formula that: d/dx K^x = In(K)*K^x
Yes, but x there isn't a function, it's the variable you're deriving in relation to! The formula for the derivate function of an exponential is:

$$\frac{d}{dx}K^{f(x)} = \frac{df(x)}{dx}ln(K)K^{f(x)}$$

The special case when f(x) = x gives that equation you wrote, because $$\frac{df(x)}{dx} = 1$$. But when f(x) isn't x, $$\frac{df(x)}{dx}$$ won't be 1.

To deduce this equation you need to do the chain rule.

LCKurtz
Homework Helper
Gold Member
The short answer is you have to use the chain rule whenever the argument isn't just a simple x. So you don't need it for ex. But you need it for e to anything more complicated than that: $e^{2x},\ e^{\sqrt x},\ e^{-x}$ etc.

It's the same with any function. You don't need it for sin(x) but you do for the sine of anything else:$\sin(2x),\ sin(-x),\ sin(x^3)$etc.

You could be safe and always use it. For example, if you use it on sin(x) you will just get the derivative as $\cos(x)\cdot 1$. The extra 1 is correct and doesn't hurt anything.

Thank you so much for all of your helps! Thanks!!

I thought you technically always use the chain rule anyway.

For instance,
$$\frac{d}{dx}y=x^2$$
$$\frac{dy}{dx}=2x(x')$$
$$\frac{dy}{dx}=2x$$

You are still using the chain rule, it's just that x' is 1 because the function is being differentiated with respect to x.