Question

How do you use the chain rule to differentiate $y = {\left( {4{x^3} - 7} \right)^4}{\left( {3x + 2} \right)^{10}}$ ?

Answer 1

Hint : We have to differentiate the given function $y$ in terms of $x$ . We can observe that the given function is a composite function. We have to use the chain rule method. Chain rule works by differentiating the outer function and then the inner function as a product. This is given as,
If $y = f\left( {g\left( x \right)} \right)$ then ${ y'} = \dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right)g'\left( x \right)$
Since the given function is a product of two expressions, we will also use the product rule.

Complete step by step solution:
We have been given the function $y = {\left( {4{x^3} - 7} \right)^4}{\left( {3x + 2} \right)^{10}}$ .
We have to use the chain rule for differentiation. The chain rule method is given for composite function as,
${ y'} = \dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right)g'\left( x \right)$
In the given function, we can observe that two expressions which are raised to some powers are multiplied to form the function. We can assume the function in two parts,
$g\left( x \right) = \left( {4{x^3} - 7} \right)$ and $h\left( x \right) = \left( {3x + 2} \right)$
Then we can write the given function as,
$y = f\left( x \right) = {\left[ {g\left( x \right)} \right]^4}{\left[ {h\left( x \right)} \right]^{10}}$
First we will use product rule to differentiate as follows,
$y' = \dfrac{{d{{\left[ {g\left( x \right)} \right]}^4}{{\left[ {h\left( x \right)} \right]}^{10}}}}{{dx}} = {\left[ {g\left( x \right)} \right]^4}\dfrac{{d\left( {{{\left[ {h\left( x \right)} \right]}^{10}}} \right)}}{{dx}} + {\left[ {h\left( x \right)} \right]^{10}}\dfrac{{d\left( {{{\left[ {g\left( x \right)} \right]}^4}} \right)}}{{dx}}$
Using chain rule for individual functions $g\left( x \right)$ and $h\left( x \right)$ we can write,
$\dfrac{{d\left( {{{\left[ {h\left( x \right)} \right]}^{10}}} \right)}}{{dx}} = 10{\left[ {h\left( x \right)} \right]^9}.h'\left( x \right) \\\ \dfrac{{d\left( {{{\left[ {g\left( x \right)} \right]}^4}} \right)}}{{dx}} = 4{\left[ {g\left( x \right)} \right]^3}.g'\left( x \right) \;$
We can find $h'\left( x \right)$ and $g'\left( x \right)$ as,
$h\left( x \right) = \left( {3x + 2} \right) \Rightarrow h'\left( x \right) = 3 \\\ g\left( x \right) = \left( {4{x^3} - 7} \right) \Rightarrow g'\left( x \right) = 12{x^2} \\\$
Now we put all the values to find the derivative of the function as,
$y' = {\left[ {g\left( x \right)} \right]^4}.10{\left[ {h\left( x \right)} \right]^9}.h'\left( x \right) + {\left[ {h\left( x \right)} \right]^{10}}.4{\left[ {g\left( x \right)} \right]^3}.g'\left( x \right) \\\ \Rightarrow y' = \left[ {{{\left( {4{x^3} - 7} \right)}^4}.10{{\left( {3x + 2} \right)}^9}.3} \right] + \left[ {{{\left( {3x + 2} \right)}^{10}}.4{{\left( {4{x^3} - 7} \right)}^3}.12{x^2}} \right] \\\ \Rightarrow y' = \left[ {30{{\left( {4{x^3} - 7} \right)}^4}{{\left( {3x + 2} \right)}^9}} \right] + \left[ {48{x^2}{{\left( {4{x^3} - 7} \right)}^3}{{\left( {3x + 2} \right)}^{10}}} \right] \\\ \Rightarrow y' = 6{\left( {4{x^3} - 7} \right)^3}{\left( {3x + 2} \right)^9}\left[ {5\left( {4{x^3} - 7} \right) + 8{x^2}\left( {3x + 2} \right)} \right] \;$
This is the derivative of the given function which is found using chain rule method.
So, the correct answer is “ $\Rightarrow y' = 6{\left( {4{x^3} - 7} \right)^3}{\left( {3x + 2} \right)^9}\left[ {5\left( {4{x^3} - 7} \right) + 8{x^2}\left( {3x + 2} \right)} \right]$ ”.

Note : We observed that the given function is composed of two expressions raised to some power and multiplied together. We used the chain rule on individual expressions after using the product rule. When using the chain rule, we should not forget to differentiate the inner function after differentiating the outer function and writing the result as the product of both.