Q.Find $\frac{dy}{dx}$ in the following: $e^x + e^{x^2} + ... + e^{x^5}$

Concept: Chain Rule — each term $e^{x^n}$ differentiates to $nx^{n-1}e^{x^n}$; sum the results. …

The pattern in the question continues through powers $x, x^2, x^3, x^4, x^5$ — five terms in total, each of the form $e^{x^n}$. Since the derivative of a sum is the sum of the derivatives, differentiate each term separately using the chain rule: for $e^{f(x)}$, the derivative is $e^{f(x)}\cdot f'(x)$.

Term 1: $e^{x}$. Inner function $x$, derivative $1$. $\dfrac{d}{dx}e^{x} = e^{x}$.

Term 2: $e^{x^2}$. Inner function $x^2$, derivative $2x$. $\dfrac{d}{dx}e^{x^2} = 2xe^{x^2}$.

Term 3: $e^{x^3}$. Inner function $x^3$, derivative $3x^2$. $\dfrac{d}{dx}e^{x^3} = 3x^2e^{x^3}$.

Term 4: $e^{x^4}$. Inner function $x^4$, derivative $4x^3$. $\dfrac{d}{dx}e^{x^4} = 4x^3e^{x^4}$. …

Method: Differentiating a Sum of Terms of the Form $e^{x^n}$

Use this whenever $y$ is a sum of several exponential terms, each with a different power of $x$ in the exponent.

Steps

Step 1: Split the sum into individual terms

Since the derivative of a sum is the sum of the derivatives, differentiate each term $e^{x^n}$ completely independently before adding the results back together.

Step 2: For each term, identify the inner function

For a term $e^{x^n}$, the outer function is $e^{u}$ and the inner function is $u=x^n$.

Step 3: Apply the chain rule to each term

$\frac{d}{dx}e^{x^n} = e^{x^n}\cdot\frac{d}{dx}(x^n) = n\,x^{n-1}\,e^{x^n}$ …

Common Mistakes

Mistake 1: Forgetting the inner-derivative factor for higher-power terms

Why it's wrong: writing $\dfrac{d}{dx}e^{x^3} = e^{x^3}$ (missing the $3x^2$ factor) is a very common slip once several similar terms are stacked in one problem. Correct approach: for every term of the form $e^{x^n}$, explicitly write the factor $n\,x^{n-1}$ before moving to the next term.

Mistake 2: Mixing up the exponent power with the coefficient

Why it's wrong: for $e^{x^4}$ the correct factor is $4x^3$ (power drops by one), not $4x^4$ or $x^3$ alone — losing either the coefficient or the reduced power gives a wrong term. Correct approach: apply the ordinary power rule to the exponent $x^n$ exactly as you would to any polynomial term, then attach that as a multiplier to $e^{x^n}$. …