Question

If \[y = {x^3} + 2x\] and \[\dfrac{{dx}}{{dt}} = 5\] , how do you find \[\dfrac{{dy}}{{dt}}\] when \[x = 2\] ?

Answer 1

Hint: Here we have a function ‘y’ dependent on variable ‘x’. We will differentiate ‘y’ with respect to ‘t’ and apply chain rule in the right hand side of the equation and differentiate the right hand side with respect to ‘t’. Substitute the values from the question in the end.
* Differentiation of\[{x^n}\]with respect to x is given by \[\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}\]
* Chain rule of differentiation:\[\dfrac{d}{{dx}}g\left( {f(x)} \right) = \dfrac{d}{{dx}}g(f(x)) \times \dfrac{d}{{dx}}f(x)\]

Complete step by step solution:
We are given the equation\[y = {x^3} + 2x\]
We differentiate both sides of the equation with respect to ‘t’.
\[ \Rightarrow \dfrac{{dy}}{{dt}} = \dfrac{d}{{dt}}\left( {{x^3} + 2x} \right)\]
Apply chain rule of differentiation on right hand side of the equation
\[ \Rightarrow \dfrac{{dy}}{{dt}} = \left( {3{x^2} + 2} \right)\dfrac{{dx}}{{dt}}\]
Now we have to calculate the value of \[\dfrac{{dy}}{{dt}}\] when \[\dfrac{{dx}}{{dt}} = 5\] and \[x = 2\] ?
Substitute these values from the question on right hand side of the equation
\[ \Rightarrow \dfrac{{dy}}{{dt}} = \left( {3{{(2)}^2} + 2} \right) \times 5\]
Multiply the terms inside the bracket on right hand side of the equation
\[ \Rightarrow \dfrac{{dy}}{{dt}} = \left( {12 + 2} \right) \times 5\]
Add the terms inside the bracket on right hand side of the equation
\[ \Rightarrow \dfrac{{dy}}{{dt}} = \left( {14} \right) \times 5\]
Calculate the product of terms on right hand side of the equation
\[ \Rightarrow \dfrac{{dy}}{{dt}} = 70\]
\[\therefore \]The value of \[\dfrac{{dy}}{{dt}} = 70\] when \[\dfrac{{dx}}{{dt}} = 5\] and \[x = 2\]

Note:
Students many times make the mistake of directly differentiating the given equation of ‘y’ in terms of ‘x’ as there is no sign of the variable ‘t’ in the equation. This tends to confuse the students and they try to forcefully create a place for variable ‘t’. Keep in mind if any third variable is introduced and we have to find the differentiation with respect to that new variable, we differentiate with respect to that variable, and apply chain rule of differentiation which will help us differentiate the function in terms of old variable and then old variable in terms of new variable.