Implicit Differentiation Step Explanation

Question

I'm trying to follow some notes on beam mechanics, and there is a differentiation step I don't understand.

He goes from:

$M_n(\lambda(1-R)+R) = \frac{1}{3} + \frac{(R-1)(1-\lambda)^2(2+\lambda)}{6}$

to:

$\frac{dM_n}{d\lambda}(\lambda(1-R)+R) + M_n(1-R) = \frac{R-1}{6}(-2(1-\lambda)(2+\lambda)+(1-\lambda)^2)$

I can do the R.H.S, but I don't understand how the L.H.S is differentiated. Also some insights into how the R.H.S is differentiated in one step would be very useful!

Can anyone help me understand this please?

Thanks.

Answer 1

Consider the lhs as a product of two functions $f(\lambda) = M_n (\lambda)$ and $g(\lambda) = \lambda(1-R) +R$.

Note that $f'(\lambda) = \frac{dM_n}{d\lambda}$, $g'(\lambda) = 1-R$.

Then the derivative wrt $\lambda$ is:

$$f'(\lambda)g(\lambda)+f(\lambda)g'(\lambda)= \frac{dM_n}{d\lambda}(\lambda(1-R) +R)+M_n (\lambda)(1-R).$$