# Questions tagged [chain-rule]

For questions involving the chain rule in analysis. The chain rule is a special rule to differentiate a composition (chain) of several functions. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule.

1,623 questions
Filter by
Sorted by
Tagged with
11k views

### Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
• 3,940
25k views

### Understanding the chain rule in probability theory

When my teacher told us about the chain rule I found it quite easy, but when I am trying to prove something based on this rule I kind of get confused about what are the allowed forms of this rule. For ...
• 313
3k views

• 73
920 views

• 1,939
6k views

### Proof of multivariable chain rule

I'm working with a proof of the multivariable chain rule $\displaystyle{\frac{d}{dt}g(t)=\frac{df}{dx_1}\frac{dx_1}{dt}+\frac{df}{dx_2}\frac{dx_2}{dt}}$ for $g(t)=f(x_1(t),x_2(t))$, but I have a hard ...
• 63
380 views

### Applying the Chain Rule for ($\sin(x) + 1$)

So, according to the chain rule, $$\frac{d(f(g(x)))}{dx} = f'(g(x)) \cdot g'(x).$$ Now, if we considered $f(x) = x+1$ and $g(x) = \sin(x)$ then: $$f(g(x)) = \sin(x)+1$$ In this case, shouldn't ...
269 views

### Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?

here's my question If $y^2=ax^2+bx+c$ then prove that $y^3[d^2y/dx^2]$ is a constant . I have solved this using the conventional method, taking square root, differentiating w.r.t to x and using ...
• 51
364 views

### Multivariable Chain Rule Formula doesn't make sense to me

Consider $z=f(x(t),y(t))$, then its chain rule formula is: $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$ What I cannot make sense of, are ...
4k views

174 views

### Is it possible to solve $f'(x)=f^{-1}(x)$?

I am interested in proving what family of functions have the property $$f'(x)=f^{-1}(x)$$ I've never dealt with a differential equation of this form, hence I could only go as far as to gather a little ...
• 4,472