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Q2:
Differentiate the following w.r.t. $x$: $e^{\sin^{-1} x}$
Solution :
Step 1: Formal Definition of the Function
Let the given function be represented as $y$:
$y = e^{\sin^{-1} x}$
[Definition of the function for differentiation]
Step 2: Identifying the Differentiation Rule
To differentiate this function, we employ the Chain Rule of calculus. Since the function is a composite function of the form $f(g(x))$, where $f(u) = e^u$ and $g(x) = \sin^{-1} x$, the derivative is given by:
$\frac{dy}{dx} = \frac{d}{du}(e^u) \cdot \frac{d}{dx}(\sin^{-1} x)$
[Per the Chain Rule for differentiation]
Step 3: Execution of the Chain Rule
Applying the standard derivative formulas for the exponential function and the inverse trigonometric function:
1. The derivative of $e^u$ with respect to $u$ is $e^u$.
2. The derivative of $\sin^{-1} x$ with respect to $x$ is $\frac{1}{\sqrt{1 - x^2}}$.
[Standard derivatives: $\frac{d}{dx}e^x = e^x$ and $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$]
Step 4: Algebraic Synthesis
Substituting the derivatives back into the Chain Rule expression:
$\frac{dy}{dx} = e^{\sin^{-1} x} \cdot \frac{d}{dx}(\sin^{-1} x)$
$\frac{dy}{dx} = e^{\sin^{-1} x} \cdot \left( \frac{1}{\sqrt{1 - x^2}} \right)$
Step 5: Visual Representation of the Differential Geometry
Step 6: Final Algebraic Refinement
Combining the terms into a single rational expression:
$\frac{dy}{dx} = \frac{e^{\sin^{-1} x}}{\sqrt{1 - x^2}}$
[Subject to the domain constraint $|x| < 1$ for the derivative to be defined]
Final Solution: $\frac{dy}{dx} = \frac{e^{\sin^{-1} x}}{\sqrt{1 - x^2}}$
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.4
- Q1: Differentiate the following w.r.t. $x$: $\frac{e^x}{\sin x}$
- 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. $x$: $e^x + e^{x^2} + ... + e^{x^5}$
- Q7: Differentiate the following w.r.t. $x$: $\sqrt{e^{\sqrt{x}}}, x > 0$
- 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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