Question

If $u(x)$ and $v(x)$ are differentiable at $x=0$, and if $u(0)=5$, $u'(0) = -3$, $v(0) = -1$ and $v'(0) = 2$, then the value of $\dfrac{d}{dx}\left(uv + \frac{u}{v}\right)$ at $x = 0$ is

• A. $-20$
• B. $-7$
• C. $6$
• D. $13$

Hint:

Always apply the product rule and quotient rule separately when both appear in the same expression.

Answer 1

The Correct Option is C

Step 1: Differentiate the expression.
Let $f(x) = uv + \dfrac{u}{v}$. Then, $f'(x) = u'v + uv' + \dfrac{u'v - uv'}{v^2}$.
Step 2: Substitute $x = 0$ values.
At $x = 0$: $u = 5$, $u' = -3$, $v = -1$, $v' = 2$.
Compute each term: $u'v = (-3)(-1) = 3$
$uv' = (5)(2) = 10$
$\dfrac{u'v - uv'}{v^2} = \dfrac{(-3)(-1) - (5)(2)}{(-1)^2} = \dfrac{3 - 10}{1} = -7$
Step 3: Add terms.
$f'(0) = 3 + 10 - 7 = 13$.
Step 4: Conclusion.
Thus, the derivative at $x=0$ is 13.