Question

If \( h(x) = \sqrt{4f(x) + 3g(x)} \), \( f(1)=4 \), \( g(1)=3 \), \( f'(1)=3 \), \( g'(1)=4 \), then \( h'(1) \) is equal to:}

• A. \( -\dfrac{5}{12} \)
• B. \( -\dfrac{12}{7} \)
• C. \( \dfrac{5}{12} \)
• D. \( \dfrac{12}{5} \)

Hint:

For functions of the form \[ h(x)=\sqrt{f(x)} \] always apply the chain rule: \[ h'(x)=\frac{f'(x)}{2\sqrt{f(x)}} \] First evaluate the inner function at the given point, then substitute the derivatives.

Answer 1

The Correct Option is C

Concept: To differentiate a composite function involving a square root, we use the chain rule. If \[ h(x) = \sqrt{u(x)} \] then \[ h'(x) = \frac{1}{2\sqrt{u(x)}} \cdot u'(x) \]
Step 1: {Identify the inner function.} \[ h(x) = \sqrt{4f(x) + 3g(x)} \] Let \[ u(x) = 4f(x) + 3g(x) \]
Step 2: {Differentiate using chain rule.} \[ h'(x) = \frac{1}{2\sqrt{4f(x)+3g(x)}} \cdot \frac{d}{dx}(4f(x)+3g(x)) \] \[ = \frac{1}{2\sqrt{4f(x)+3g(x)}} (4f'(x)+3g'(x)) \]
Step 3: {Substitute the given values at \(x=1\).} First evaluate the expression inside the root: \[ 4f(1)+3g(1) = 4(4) + 3(3) \] \[ = 16 + 9 = 25 \] Thus, \[ \sqrt{25} = 5 \] Now compute the derivative term: \[ 4f'(1) + 3g'(1) = 4(3) + 3(4) \] \[ = 12 + 12 = 24 \]
Step 4: {Calculate \(h'(1)\).} \[ h'(1) = \frac{1}{2 \times 5} \times 24 \] \[ = \frac{24}{10} \] \[ = \frac{12}{5} \] Hence, \[ h'(1) = \frac{12}{5} \] Thus, the correct option is (D).