Why Is the Chain Rule Necessary for Differentiating Functions Like e^sqrt(x)?

[sl2382]

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!

 

[Mark44]

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(e^x) = e^x, but what about d/dx(e^f(x))? For that you need the chain rule.

Let u = f(x).
d/dx(e^f(x)) = d/dx(e^u) = d/du(e^u) * du/dx = e^u * du/dx.

 

[Tosh5457]

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]

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 e^x. 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.

 

[sl2382]

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

 

[QuarkCharmer]

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.