prior to goinns come discover ns derivative the e2x, lens uns recall a few facts about ns exponentiatogether functions. In math, exponential features are the the develop f(x) = ax, where 'a' is a consistent and also 'x' ins a variable. Here, the consistent 'a' have to be greater 보다 0 for f(x) to be one exponential function. Part various other forms the exponentiatogether functions to be abx, abkx, ex, pekx, and so on Thus, e2x ins additionally an exponentiatogether attribute and also ns derivati have of e2x is 2e2x.

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us are going to find ns derivati have of e2x in different approaches and we will certainly additionally settle a couple of Instances using the same.

1. | What ins ns Derivative of e^2x? |

2. | Derivative the e^2x evidence by first Principle |

3. | Derivative that e^2x evidence by Chain Rule |

4. | Derivati have the e^2x proof through Logarithmic Differentiation |

5. | n^th Derivative that e^2x |

6. | faqs on Derivative of e^2x |

## Wcap ins the Derivati have of e^2x?

ns derivative of e2x with respecns come x ins 2e2x. We write this mathematicallied together **d/dx (e2x) = 2e2x (or) (e2x)' = 2e2x**. Here, f(x) = e2x is one exponentiatogether function as the base ins 'e' is a constant (which ins well-known together Euler's number and also itns worth ins about 2.718) and also the border recipe the 'e' ins lim ₙ→∞ (1 + (1/n))n. We can execute ns differentiatitop top of e2x in different approaches together as:

### Derivative that e^2x Formula

ns derivative that e2x is 2e2x. It have the right to be created as

**d/dx (e2x) = 2e2x**(or)

**(e2x)' = 2e2x**

Let us prove thins in different techniques together discussed above.

## Derivative that e^2x proof through initially Principle

below ins ns differentiatitop top that e2x by the initially principle. For this, lens uns assume that f(x) = e2x. Then f(x + h) = e2(x + h) = e2x + 2h. Substituting these values in ns formula that the derivati have utilizing initially principle (i m sorry ins additionally well-known together ns border interpretation that the derivative),

f'(x) = limₕ→₀

f'(x) = limₕ→₀

= limₕ→₀

= limₕ→₀

= e2x limₕ→₀ (e2h - 1) / h

assume that 2h = t. Climate together h → 0, 2h → 0. I.e., ns → 0 together well. Then ns above limit becomes,

= e2x limₜ→₀ (ens - 1) / (t / 2)

= 2e2x limₜ→₀ (ens - 1) / t

utilizing border formulas, us have limₜ→₀ (et - 1) / ns = 1. So

f'(x) = 2e2x (1) = 2e2x

Thus, ns derivati have the e2x ins uncovered by ns first principle.

## Derivative of e^2x proof by Chain Rule

us have the right to execute ns differentiati~ above the e2x utilizing ns chain dominance Due to the fact that e2x can it is in to express together a compowebsite function. I.e., us have the right to write e2x = f(g(x)) where f(x) = ex and g(x) = 2x (one deserve to conveniently verify the f(g(x)) = e2x).

then f'(x) = ex and also g'(x) = 2. By chain rule, the derivative that f(g(x)) is f'(g(x)) · g'(x). Making use of this,

d/dx (e2x) = f'(g(x)) · g'(x)

= f'(2x) · (2)

= e2x (2)

= 2e2x

Thus, the derivative that e2x is uncovered by making use of the chain rule.

## Derivati have that e^2x proof through Logarithmic Differentiation

us understand the the logarithmic differentiation ins supplied come distinguish a exponential feature and also thus ins have the right to be used come find ns derivative of e2x. For this, let us assume the y = e2x. As a procedure the logarithmic differentiation, us take it ns natural logarithm (ln) top top both political parties of the above equation. Then us get

ln y = ln e2x

among the nature that logarithmns ins ln am = m ln a. Using this,

ln y = 2x ln e

us understand the ln e = 1. So

ln y = 2x

distinguishing both political parties via respect come x,

(1/y) (dy/dx) = 2(1)

dy/dx = 2y

Substitutinns y = e2x ago here,

d/dx(e2x) = 2e2x

Thus, us have discovered the derivati have that e2x through making use of logarithmic differentiation.

## n^th Derivative the e^2x

nthderivati have of e2x is ns derivative of e2x the ins acquired through differentiating e2x consistently for n times. To find the nthderivati have that e2xx, let us discover ns first derivative, 2nd derivative, ... Up to a couple of time come understand ns trend.

1stderivative the e2x ins 2e2x2ndderivati have that e2x ins 4e2x3rdderivati have the e2x is 8e2x4thderivati have the e2x ins 16 e2xand so on.Thus, ns nthderivative ofe2xis:

**dn/(dxn) (e2x) = 2ne2x**

**important notes top top Derivati have of** **e2x:**

**Topics pertained to Derivative the e2x:**

## Examples utilizing Derivati have of e^2x

**example 1:** discover ns derivati have that e2x + 1.

**Solution:**

Lens f(x) = e2x + 1

utilizing ns chain rule,

f'(x) = e2x + 1 d/dx (2x + 1)

= e2x + 1 (2)

= 2e2x + 1

**Answer:** ns derivative that e2x + 1 is 2e2x + 1.

**instance 2:** Wcap ins ns derivative that e2x + e-2x?

**Solution:**

Let f(x) = e2x + e-2x

making use of ns chain rule,

f'(x) = e2x · d/dx (2x) + e-2x · d/dx (-2x)

= e2x (2) + e-2x (-2)

= 2 (e2x - e-2x)

**Answer:** the derivative that e2x + e-2x ins 2 (e2x - e-2x).

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## frequently asked questions top top Derivative the e^2x

### Wcap ins Derivati have that e2x?

ns derivati have the e2x is 2e2x. Mathematically, ins ins created together d/dx(e2x) = 2e2x (or) (e2x)' = 2e2x.

### just how to identify e come the power of 2x?

Let f(x) = e2x. By using chain rule, ns derivative that e2x is, e2x d/dx (2x) = e2x (2) = 2 e2x. Thus, ns derivati have that e to ns strength the 2x is 2e2x.

### What is the Derivative the e3x?

Let f(x) = e3x. By using chain rule, the derivative the e3x is, e3x d/dx (3x) = e3x (2) = 3 e3x. Thus, the derivati have of e3x ins 3e3x.

### how come find the Derivati have that e2x + 3?

Lens us assume the f(x) = e2x + 3. Utilizing ns chain rule, f'(x) = e2x + 3 d/dx (2x + 1) = e2x + 3 (2) = 2e2x + 3. Thus, the derivati have of e2x + 3 ins 2e2x + 3.

### Ins the Derivati have that e2x very same together ns Integral that e2x?

No, ns derivati have the e2x ins not exact same together the integral that e2x.

the derivative that e2x is 2e2x.ns integral the e2x ins e2x / 2.### What ins the Derivative that e2x²?

Lens f(x) = e2x². By ns applications that chain rule, f'(x) = e2x² d/dx (2x2) = e2x² (4x) = 4x e2x². Thus, ns derivati have of e2x² is 4x e2x².

### just how to uncover the Derivati have that e2x by first Principle?

Lens f(x) = e2x. By initially principle, f'(x) = limₕ→₀

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### Wcap ins the Derivative of e2x sin 3x?

Lens f(x) = e2x sin 3x. By producns rule, f'(x) = e2x d/dx (sin 3x) + sin 3x d/dx (e2x) = e2x (cons 3x) d/dx (3x) + sin 3x (2e2x) = e2x (3 cons 3x + 2 sin 3x). Thus, the derivative that e2x sin 3x ins e2x (3 cons 3x + 2 sin 3x).