Rivalry2

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Original 2026-01-01

Modified 2026-02-28

1 We can derive the derivative of 6^x using proofs.

2 To show this, we will use the properties of exponential functions along with the rules of differentiation.

3 There are several methods we use to prove this, such as:

4 By First Principle

5 Using Chain Rule

6 Using Logarithmic Differentiation

7 We will now demonstrate that the differentiation of 6^x results in 6^x ln(6) using the above-mentioned methods:

8 By First Principle

9 The derivative of \(6^x\) can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.

10 To find the derivative of\( 6^x\) using the first principle, we will consider f(x) =\( 6^x\).

11 Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

12 Given that f(x) = \(6^x\), we write f(x + h) = \(6^(x + h)\).

13 Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [\(6^(x + h) - 6^x\)] / h = limₕ→₀ [\(6^x (6^h - 1)\)] / h = \(6^x\) limₕ→₀ [\((6^h - 1) / h\)]

14 Using the limit property, limₕ→₀ [\((6^h - 1) / h\)] = ln(6). f'(x) = \(6^x\) ln(6)

15 Hence, proved.

16 Using Chain Rule

17 To prove the differentiation of 6^x using the chain rule, We express 6^x as e^(x ln(6)).

18 Let u = x ln(6). Then, 6^x = e^u. By the chain rule: d/dx [e^u] = e^u (du/dx)

19 So we get, d/dx (6^x) = e^(x ln(6)) * ln(6)

20 Substituting back the<a>expression</a>for e^(x ln(6)) gives us: d/dx (6^x) = 6^x ln(6).

21 Using Logarithmic Differentiation

22 We will now prove the derivative of \(6^x \)using logarithmic differentiation.

23 The step-by-step process is demonstrated below: Consider y = \(6^x\)

24 Taking the natural logarithm of both sides gives ln(y) = ln(\(6^x\)) = x ln(6)

25 Differentiating both sides with respect to x gives: (1/y) dy/dx = ln(6)

26 Thus, dy/dx = y ln(6) = \(6^x \)ln(6)

27 Therefore, the derivative of the function y =\( 6^x\) is \(6^x\) ln(6).