The solutions here require the use of the 'chain rule'.
\nCertain functions often define the input variable $x$ as being within another function (a function of a function).
\nFor example, $y=2(3x^3+1)^2$
\nHere, an inner function is shown in brackets as: $3x^3+1$, which in turn is squared and then multiplied by two: $2(3x^3+1)^2$
\nTo differentiate such a function, we can use the chain rule, which states:
\n$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}$
\nThis works by first defining the inner function as $u$. In this case: $u=3x^3+1$. Notice that this forms a new equation where $u$ can be differentiated in terms of $x$ using the power rule: $\\frac{du}{dx}=(3\\times3)x^{3-1}+(1\\times0)x^{0-1}=9x^2+0=9x^2$
\nUsing the $u$-substitution, the original equation becomes $y=2u^2$ and $y$ can now be differentiated with respect to $u$ with the power rule: $\\frac{dy}{du}=(2\\times2)u^{2-1}=4u^1=4u$
\nWe've now found both $\\frac{dy}{du}$ and $\\frac{du}{dx}$
\n$\\therefore\\;\\;\\;\\frac{dy}{dx}=4u\\times9x^2=36x^2u$
\nThe original expression for $u$ is then substiuted back in to give the final answer:
\n$\\frac{dy}{dx}=36x^2(3x+1)$ this could be further expanded to give $\\frac{dy}{dx}=108x^3+36x^2$
\nWith practice, as with many other mathematical processes, differentiating in this way can be achieved by 'inspection', where you mentally think ahead and carry out the steps without writing down the $u$ subsitution. It's always advisable to train yourself to use the full process first and move onto inspection when you're ready.
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\n$\\frac{dy}{dx}=$ [[0]]
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\n$\\frac{dy}{dx}=$ [[0]]
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\nDo not write out $dy/dx$; only input the differentiated right hand side of each equation.
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