Answer: The correct option is (B) f(q) = 2q + 3.
Step-by-step explanation: The given equation representing a function is
We are to write the equation in function notation with 'q' as independent variable.
From equation (i), we have
So, 's' can be represented as a function of 'q'. Hence, we can write
Therefore, we have
Thus, (B) is the correct option.
Differentiate both sides of the equation.<span><span><span>d<span>dx</span></span><span>(<span>x3</span>+<span>y3</span>)</span>=<span>d<span>dx</span></span><span>(18xy)</span></span><span><span>d<span>dx</span></span><span>(<span>x3</span>+<span>y3</span>)</span>=<span>d<span>dx</span></span><span>(18xy)</span></span></span>Differentiate the left side of the equation.Tap for fewer steps...By the Sum Rule, the derivative of <span><span><span>x3</span>+<span>y3</span></span><span><span>x3</span>+<span>y3</span></span></span> with respect to <span>xx</span> is <span><span><span>d<span>dx</span></span><span>[<span>x3</span>]</span>+<span>d<span>dx</span></span><span>[<span>y3</span>]</span></span><span><span>d<span>dx</span></span><span>[<span>x3</span>]</span>+<span>d<span>dx</span></span><span>[<span>y3</span>]</span></span></span>.<span><span><span>d<span>dx</span></span><span>[<span>x3</span>]</span>+<span>d<span>dx</span></span><span>[<span>y3</span>]</span></span><span><span>d<span>dx</span></span><span>[<span>x3</span>]</span>+<span>d<span>dx</span></span><span>[<span>y3</span>]</span></span></span>Differentiate using the Power Rule which states that <span><span><span>d<span>dx</span></span><span>[<span>xn</span>]</span></span><span><span>d<span>dx</span></span><span>[<span>xn</span>]</span></span></span> is <span><span>n<span>x<span>n−1</span></span></span><span>n<span>x<span>n-1</span></span></span></span> where <span><span>n=3</span><span>n=3</span></span>.<span><span>3<span>x2</span>+<span>d<span>dx</span></span><span>[<span>y3</span>]</span></span><span>3<span>x2</span>+<span>d<span>dx</span></span><span>[<span>y3</span>]</span></span></span>Evaluate <span><span><span>d<span>dx</span></span><span>[<span>y3</span>]</span></span><span><span>d<span>dx</span></span><span>[<span>y3</span>]</span></span></span>.Tap for more steps...<span><span>3<span>x2</span>+3<span>y2</span><span>d<span>dx</span></span><span>[y]</span></span><span>3<span>x2</span>+3<span>y2</span><span>d<span>dx</span></span><span>[y]</span></span></span>Differentiate the right side of the equation.Tap for fewer steps...Since <span>1818</span> is constant with respect to <span>xx</span>, the derivative of <span><span>18xy</span><span>18xy</span></span> with respect to <span>xx</span> is <span><span>18<span>d<span>dx</span></span><span>[xy]</span></span><span>18<span>d<span>dx</span></span><span>[xy]</span></span></span>.<span><span>18<span>d<span>dx</span></span><span>[xy]</span></span><span>18<span>d<span>dx</span></span><span>[xy]</span></span></span>Differentiate using the Product Rule which states that <span><span><span>d<span>dx</span></span><span>[f<span>(x)</span>g<span>(x)</span>]</span></span><span><span>d<span>dx</span></span><span>[f<span>(x)</span>g<span>(x)</span>]</span></span></span> is <span><span>f<span>(x)</span><span>d<span>dx</span></span><span>[g<span>(x)</span>]</span>+g<span>(x)</span><span>d<span>dx</span></span><span>[f<span>(x)</span>]</span></span><span>f<span>(x)</span><span>d<span>dx</span></span><span>[g<span>(x)</span>]</span>+g<span>(x)</span><span>d<span>dx</span></span><span>[f<span>(x)</span>]</span></span></span> where <span><span>f<span>(x)</span>=x</span><span>f<span>(x)</span>=x</span></span> and <span><span>g<span>(x)</span>=y</span><span>g<span>(x)</span>=y</span></span>.<span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y<span>d<span>dx</span></span><span>[x]</span>)</span></span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y<span>d<span>dx</span></span><span>[x]</span>)</span></span></span>Rewrite <span><span><span>d<span>dx</span></span><span>[y]</span></span><span><span>d<span>dx</span></span><span>[y]</span></span></span> as <span><span><span>d<span>dx</span></span><span>[y]</span></span><span><span>d<span>dx</span></span><span>[y]</span></span></span>.<span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y<span>d<span>dx</span></span><span>[x]</span>)</span></span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y<span>d<span>dx</span></span><span>[x]</span>)</span></span></span>Differentiate using the Power Rule which states that <span><span><span>d<span>dx</span></span><span>[<span>xn</span>]</span></span><span><span>d<span>dx</span></span><span>[<span>xn</span>]</span></span></span> is <span><span>n<span>x<span>n−1</span></span></span><span>n<span>x<span>n-1</span></span></span></span> where <span><span>n=1</span><span>n=1</span></span>.<span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y⋅1)</span></span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y⋅1)</span></span></span>Multiply <span>yy</span> by <span>11</span> to get <span>yy</span>.<span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y)</span></span><span>18<span>(x<span>d<span>dx</span></span><span>[y]</span>+y)</span></span></span>Simplify.Tap for more steps...<span><span>18x<span>d<span>dx</span></span><span>[y]</span>+18y</span><span>18x<span>d<span>dx</span></span><span>[y]</span>+18y</span></span>Reform the equation by setting the left side equal to the right side.<span><span>3<span>x2</span>+3<span>y2</span>y'=18xy'+18y</span><span>3<span>x2</span>+3<span>y2</span>y′=18xy′+18y</span></span>Since <span><span>18xy'</span><span>18xy′</span></span> contains the variable to solve for, move it to the left side of the equation by subtracting <span><span>18xy'</span><span>18xy′</span></span> from both sides.<span><span>3<span>x2</span>+3<span>y2</span>y'−18xy'=18y</span><span>3<span>x2</span>+3<span>y2</span>y′-18xy′=18y</span></span>Since <span><span>3<span>x2</span></span><span>3<span>x2</span></span></span> does not contain the variable to solve for, move it to the right side of the equation by subtracting <span><span>3<span>x2</span></span><span>3<span>x2</span></span></span> from both sides.<span><span>3<span>y2</span>y'−18xy'=−3<span>x2</span>+18y</span><span>3<span>y2</span>y′-18xy′=-3<span>x2</span>+18y</span></span>Factor <span><span>3y'</span><span>3y′</span></span> out of <span><span>3<span>y2</span>y'−18xy'</span><span>3<span>y2</span>y′-18xy′</span></span>.Tap for fewer steps...Factor <span><span>3y'</span><span>3y′</span></span> out of <span><span>3<span>y2</span>y'</span><span>3<span>y2</span>y′</span></span>.<span><span>3y'<span>(<span>y2</span>)</span>−18xy'=−3<span>x2</span>+18y</span><span>3y′<span>(<span>y2</span>)</span>-18xy′=-3<span>x2</span>+18y</span></span>Factor <span><span>3y'</span><span>3y′</span></span> out of <span><span>−18xy'</span><span>-18xy′</span></span>.<span><span>3y'<span>(<span>y2</span>)</span>+3y'<span>(−6x)</span>=−3<span>x2</span>+18y</span><span>3y′<span>(<span>y2</span>)</span>+3y′<span>(-6x)</span>=-3<span>x2</span>+18y</span></span>Factor <span><span>3y'</span><span>3y′</span></span> out of <span><span>3y'<span>y2</span>+3y'<span>(−6x)</span></span><span>3y′<span>y2</span>+3y′<span>(-6x)</span></span></span>.<span><span>3y'<span>(<span>y2</span>−6x)</span>=−3<span>x2</span>+18y</span><span>3y′<span>(<span>y2</span>-6x)</span>=-3<span>x2</span>+18y</span></span>Divide each term by <span><span><span>y2</span>−6x</span><span><span>y2</span>-6x</span></span> and simplify.Tap for fewer steps...Divide each term in <span><span>3y'<span>(<span>y2</span>−6x)</span>=−3<span>x2</span>+18y</span><span>3y′<span>(<span>y2</span>-6x)</span>=-3<span>x2</span>+18y</span></span> by <span><span><span>y2</span>−6x</span><span><span>y2</span>-6x</span></span>.<span><span><span><span>3y'<span>(<span>y2</span>−6x)</span></span><span><span>y2</span>−6x</span></span>=−<span><span>3<span>x2</span></span><span><span>y2</span>−6x</span></span>+<span><span>18y</span><span><span>y2</span>−6x</span></span></span><span><span><span>3y′<span>(<span>y2</span>-6x)</span></span><span><span>y2</span>-6x</span></span>=-<span><span>3<span>x2</span></span><span><span>y2</span>-6x</span></span>+<span><span>18y</span><span><span>y2</span>-6x</span></span></span></span>Reduce the expression by cancelling the common factors.Tap for more steps...<span><span>3y'=−<span><span>3<span>x2</span></span><span><span>y2</span>−6x</span></span>+<span><span>18y</span><span><span>y2</span>−6x</span></span></span><span>3y′=-<span><span>3<span>x2</span></span><span><span>y2</span>-6x</span></span>+<span><span>18y</span><span><span>y2</span>-6x</span></span></span></span>Simplify the right side of the equation.Tap for more steps...<span><span>3y'=−<span><span>3<span>(<span>x2</span>−6y)</span></span><span><span>y2</span>−6x</span></span></span><span>3y′=-<span><span>3<span>(<span>x2</span>-6y)</span></span><span><span>y2</span>-6x</span></span></span></span>Divide each term by <span>33</span> and simplify.Tap for fewer steps...Divide each term in <span><span>3y'=−<span><span>3<span>(<span>x2</span>−6y)</span></span><span><span>y2</span>−6x</span></span></span><span>3y′=-<span><span>3<span>(<span>x2</span>-6y)</span></span><span><span>y2</span>-6x</span></span></span></span> by <span>33</span>.<span><span><span><span>3y'</span>3</span>=−<span><span><span>3<span>(<span>x2</span>−6y)</span></span><span><span>y2</span>−6x</span></span>3</span></span><span><span><span>3y′</span>3</span>=-<span><span><span>3<span>(<span>x2</span>-6y)</span></span><span><span>y2</span>-6x</span></span>3</span></span></span>Reduce the expression by cancelling the common factors.Tap for more steps...<span><span>y'=−<span><span><span>3<span>(<span>x2</span>−6y)</span></span><span><span>y2</span>−6x</span></span>3</span></span><span>y′=-<span><span><span>3<span>(<span>x2</span>-6y)</span></span><span><span>y2</span>-6x</span></span>3</span></span></span>Simplify the right side of the equation.Tap for more steps...<span><span>y'=−<span><span><span>x2</span>−6y</span><span><span>y2</span>−6x</span></span></span><span>y′=-<span><span><span>x2</span>-6y</span><span><span>y2</span>-6x</span></span></span></span>Replace <span><span>y'</span><span>y′</span></span> with <span><span><span>dy</span><span>dx</span></span><span><span>dy</span><span>dx</span></span></span>.<span><span><span>dy</span><span>dx</span></span>=−<span><span><span><span>x2</span>−6y</span><span><span>y2</span>−6x</span></span></span></span>
The exponential function has a steeper slope than that of the linear function, therefore:
<span>B The exponential function is generally growing faster than the linear function.</span>
