Can someone explain the chain rule?

Question

Scarlett Allison · Accepted Answer

Please see below.

We may have done different kinds of differentiation when a function is say #f(x)=x^3+x^2+2# or #g(x)=tanx# or #h(x)=sqrtx# or #t(x)=lnx# or #u(x)=e^x# or similar functions. But how does one differentiate when we have a combination of functions such as #sqrt(x^3+x^2+2)# or #ln(tanx)# or #e^sqrtx#. Observe that we can write them as #h(f(x))# or #t(g(x))# or #u(h(x))#. Hence such functions are called function of a function. Sometimes it could be more complicated i.e. function of a function of a function. For example, let us say #ln[tan(sqrt(x^3+x^2+2))]# which is just #t(g(f(x)))#. How do we deal with such function of a function of another function? In order to differentiate such complicated functions, we need to use Chain Rule, according to when we have #y=y(u(x))# then #(dy)/(dx)=(dy)/(du)xx(du)/(dx)# For example, for #y=sqrt(x^3+x^2+2)#, where #f(x)=sqrtx# and #g(x)=x^3+x^2+2#, then #y=f(g(x))# and #(dy)/(dx)=(df)/(dg)xx(dg)/(dx)# As #f=sqrtg#, we have #(df)/(dg)=1/(2sqrtg)# and as #g=x^3+x^2+2#, #(dg)/(dx)=3x^2+2x# and #(dy)/(dx)=1/(2sqrtg)xx(3x^3+2x)=(3x^3+2x)/(2sqrt(x^3+x^2+2)# as #g=x^3+x^2+2# Similarly when we have #y=f(g(h(x)))#, we have #(dy)/(dx)=(dy)/(df)xx(df)/(dg)xx(dg)/(dh)# For example for #y=ln[tan(sqrt(x^3+x^2+2))]# #(dy)/(dx)=1/tan(sqrt(x^3+x^2+2))xxsec^2(sqrt(x^3+x^2+2))xx1/(2sqrt(x^3+x^2+2)xx(3x^2+2x)#

Charles Arocha · Answer

See below.

First, I assume you mean the chain rule. Applied when we want to find the derivative of a function of a function of the variable. Let's assume that we have a function #F(x)# defined as #F(x) = f(g(x))# where #f# and #g# are both functions of #x# Then the chain rule states that: #F'(x) = f'(g(x)) * g'(x)# As an example, let's consider #y= e^(2x)# and we want to find the first derivative, #dy/dx# Using the nomenclature above: #F(x)=y, f(x) = e^x and g(x) = 2x# Then note that: #F'(x) = dy/dx, f'(x) = e^x and g'(x) =2# Hence, applying the chain rule: #dy/dx = e^(2x)*2 = 2e^(2x)# I hope this helps. [After a little practice you'll find you will be able to apply this rule in much more complicated cases without needing to go back to the definition.]