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Q6:
Differentiate the following w.r.t. $x$: $e^x + e^{x^2} + ... + e^{x^5}$
Solution :
Step 1: Formal Representation of the Function
Let the given function be represented by $y(x)$:
$y = e^x + e^{x^2} + e^{x^3} + e^{x^4} + e^{x^5}$
Step 2: Applying the Linearity Property of Differentiation
To find the derivative $\frac{dy}{dx}$, we apply the Sum Rule of differentiation, which states that the derivative of a sum of functions is the sum of their individual derivatives [Per the Linearity of the Differentiation Operator]:
$\frac{dy}{dx} = \frac{d}{dx}(e^x) + \frac{d}{dx}(e^{x^2}) + \frac{d}{dx}(e^{x^3}) + \frac{d}{dx}(e^{x^4}) + \frac{d}{dx}(e^{x^5})$
Step 3: Application of the Chain Rule
For composite functions of the form $e^{f(x)}$, we apply the Chain Rule [By the Chain Rule: $\frac{d}{dx}(e^{u}) = e^u \cdot \frac{du}{dx}$]:
- $\frac{d}{dx}(e^x) = e^x \cdot \frac{d}{dx}(x) = e^x \cdot 1 = e^x$
- $\frac{d}{dx}(e^{x^2}) = e^{x^2} \cdot \frac{d}{dx}(x^2) = e^{x^2} \cdot (2x) = 2x e^{x^2}$
- $\frac{d}{dx}(e^{x^3}) = e^{x^3} \cdot \frac{d}{dx}(x^3) = e^{x^3} \cdot (3x^2) = 3x^2 e^{x^3}$
- $\frac{d}{dx}(e^{x^4}) = e^{x^4} \cdot \frac{d}{dx}(x^4) = e^{x^4} \cdot (4x^3) = 4x^3 e^{x^4}$
- $\frac{d}{dx}(e^{x^5}) = e^{x^5} \cdot \frac{d}{dx}(x^5) = e^{x^5} \cdot (5x^4) = 5x^4 e^{x^5}$
Step 4: Summation of Individual Derivatives
Substituting the results from Step 3 back into the summation expression:
$\frac{dy}{dx} = e^x + 2x e^{x^2} + 3x^2 e^{x^3} + 4x^3 e^{x^4} + 5x^4 e^{x^5}$
Visual Representation of the Power Gradient
Final Solution: $\frac{dy}{dx} = e^x + 2x e^{x^2} + 3x^2 e^{x^3} + 4x^3 e^{x^4} + 5x^4 e^{x^5}$
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. $x$: $\log(\cos e^x)$
- 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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