We can use the chain rule to differentiate, which states we will need to multiply the derivative of the outside function by the derivative of the inside function. We find the derivative of the inside function, \(\displaystyle 3x\), to be \(\displaystyle 3\). The derivative of the outside function \(\displaystyle \cos(x)\), will be \(\displaystyle -\sin(x)\). Multiplying these values together results in \(\displaystyle -3\sin(3x)\).

To find the differential, take the derivative of each term as follows.

The derivative of anything in the form of \(\displaystyle x^n\) is \(\displaystyle n{x^{n-1}}\) so applying that rule to all of the terms yields:

 \(\displaystyle dy=(3x^2+2)dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of \(\displaystyle sin(x)\) is \(\displaystyle cos(x)\) and anything in the form of \(\displaystyle x^n\) is \(\displaystyle n{x^{n-1}}\), so applying that rule to all of the terms yields:

 \(\displaystyle dy=(cosx +2x)dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative of each term as you apply the quotient rule.

The quotient rule is:

 \(\displaystyle \frac{(denominator)(d(numerator))-(numerator)(d(denominator))}{denominator^2}\),

so applying that rule to the equation yields: 

\(\displaystyle dy=\frac{{cos(x)(2x)-(x^2)(-sinx)}}{{cos^2(x)}}dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative as follows.

The derivative of anything in the form of \(\displaystyle x^n\)is \(\displaystyle n{x^{n-1}}\), so applying that rule to all of the terms yields:

 \(\displaystyle dy=(30x)dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of \(\displaystyle tan(x)\) is \(\displaystyle sec^2(x)\), and derivative of anything in the form of \(\displaystyle x^n\) is \(\displaystyle n{x^{n-1}}\), so applying that rule to all of the terms yields:

 \(\displaystyle dy=(sec^2(x)+2x)dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative of each term as you apply the product rule.

The product rule is

\(\displaystyle (second\quad term)\cdot d(first\quad term) + (first\quad term)\cdot d(second\quad term)\), so applying that rule to the equation yields: 

\(\displaystyle dy=(sec^2(x)sin(x)+cos(x)tan(x))dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative of each term as you apply the product rule.

The product rule is: 

\(\displaystyle (second\quad term)\cdot d(first\quad term) + (first\quad term)\cdot d(second\quad term)\), so applying that rule to the equation yields: 

\(\displaystyle dy=\left(\frac{e^x}{x}+ln(x)e^x\right)dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of anything in the form of \(\displaystyle x^n\) is \(\displaystyle n{x^{n-1}}\), and the derivative of \(\displaystyle e^x\) is \(\displaystyle e^x\)so applying that rule to all of the terms yields [correct answer]:

\(\displaystyle dy=(2x+e^x)dx\)

The differential of \(\displaystyle y\) is \(\displaystyle dy\).

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of \(\displaystyle e^{-x}\)is \(\displaystyle -e^{-x}\), and the derivative of \(\displaystyle sin(x)\) is \(\displaystyle cos(x)\), so applying that rule to all of the terms yields [correct answer]:

\(\displaystyle dy=(-e^{-x}+cos(x))dx\)