Question

If $y = f({x^2} + 2)\;$ and $f\prime (3) = 5$, then $\dfrac{{dy}}{{dx}}$ at $x = 1$ is
A. $5$
B. $25$
C. $15$
D. $10$

Answer 1

Hint: Since the problem is based on differentiation of a function and we know that $f'(x)$ represents first derivative of a function with respect to $x$ hence, we will first calculate $f'(x)$ of a given function in the problem and then calculate the answer by substituting the value of $x = 1$ in the last step of the solution.

Formula used:
The formula used in this problem is the derivative of ${x^n}$ which is given as:
$\dfrac{d}{{dx}}({x^n}) = n.{x^{n - 1}}$

Complete Step by Step Solution:
The function given in the question is $y = f({x^2} + 2)\,\,\,\,\,\,\,\,\,...\,(1)$
We know that, $\dfrac{d}{{dx}}({x^n}) = n.{x^{n - 1}}$
On differentiating equation $(1)$ with respect to $x$ on both sides, we get
$\dfrac{{dy}}{{dx}} = f'({x^2} + 2)\, \cdot 2x$
Now, substitute the value $x = 1$ in the above expression, we get
${\left( {\dfrac{{dy}}{{dx}}} \right)_{x = 1}} = f'\left( {{{(1)}^2} + 2} \right) \cdot 2\left( 1 \right) = f'\left( 3 \right) \cdot 2\,\,\,\,\,\,\,\,\,\,...\,(2)$
But on substituting the value of $f'\left( 3 \right) = 5$ in equation $(2)$, we get
${\left( {\dfrac{{dy}}{{dx}}} \right)_{x = 1}} = 5 \cdot 2 = 10$
Thus, the value of $\dfrac{{dy}}{{dx}}$ at $x = 1$ is $10$.
Hence, the correct option is (D) $10$ .

Note: In this question, we have to differentiate the given equation with respect to $x$ by applying the required formula as a result of which the value of $f'$ is obtained. The value of $f'$ is given in the question, after putting its value we will get an accurate answer of the given problem.