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Q7:
Differentiate the following w.r.t. $x$: $\sqrt{e^{\sqrt{x}}}, x > 0$
Solution :
Step 1: Identifying the Function and Applying Chain Rule
Let the given function be defined as:
$y = \sqrt{e^{\sqrt{x}}}$
Rewrite the function using exponential notation to facilitate differentiation:
$y = (e^{\sqrt{x}})^{1/2} = e^{\frac{1}{2}\sqrt{x}}$ [By Law of Exponents: $(a^m)^n = a^{mn}$]
Step 2: Applying the Chain Rule of Differentiation
To differentiate $y$ with respect to $x$, we apply the Chain Rule, which states that for a composite function $f(g(x))$, the derivative is $f'(g(x)) \cdot g'(x)$. [Per the Chain Rule of Calculus].
Let $u = \frac{1}{2}\sqrt{x} = \frac{1}{2}x^{1/2}$. Then $y = e^u$.
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
Step 3: Calculating Derivative Components
Part A: Differentiation of the exponential term
$\frac{d}{du}(e^u) = e^u$ [Standard derivative formula for exponential functions].
Substituting $u$ back, we get: $e^{\frac{1}{2}\sqrt{x}}$.
Part B: Differentiation of the exponent term
$\frac{d}{dx}\left(\frac{1}{2}x^{1/2}\right) = \frac{1}{2} \cdot \frac{d}{dx}(x^{1/2})$
Using the Power Rule $\frac{d}{dx}(x^n) = nx^{n-1}$:
$\frac{1}{2} \cdot \left(\frac{1}{2}x^{1/2 - 1}\right) = \frac{1}{4}x^{-1/2} = \frac{1}{4\sqrt{x}}$
Step 4: Combining the Components
Multiplying the results from Part A and Part B:
$\frac{dy}{dx} = e^{\frac{1}{2}\sqrt{x}} \cdot \frac{1}{4\sqrt{x}}$
Substitute $e^{\frac{1}{2}\sqrt{x}}$ back to the original radical form $\sqrt{e^{\sqrt{x}}}$:
$\frac{dy}{dx} = \frac{\sqrt{e^{\sqrt{x}}}}{4\sqrt{x}}$
Final Solution: $\frac{dy}{dx} = \frac{\sqrt{e^{\sqrt{x}}}}{4\sqrt{x}}$
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.4
- Q1: Differentiate the following w.r.t. $x$: $\frac{e^x}{\sin x}$
- Q2: Differentiate the following w.r.t. $x$: $e^{\sin^{-1} x}$
- Q3: Differentiate the following w.r.t. $x$: $e^{x^3}$
- Q4: Differentiate the following w.r.t. $x$: $\sin(\tan^{-1} e^{-x})$
- Q5: Differentiate the following w.r.t. $x$: $\log(\cos e^x)$
- Q6: Differentiate the following w.r.t. $x$: $e^x + e^{x^2} + ... + e^{x^5}$
- Q8: Differentiate the following w.r.t. $x$: $\log(\log x), x > 1$
- Q9: Differentiate the following w.r.t. $x$: $\frac{\cos x}{\log x}, x > 0$
- Q10: Differentiate the following w.r.t. $x$: $\cos(\log x + e^x), x > 0$
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.1
- Continuity and Differentiability Class 12 EXERCISE 5.2
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
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