Questions: 2. Complete the following derivatives. (a) The derivative of e^x with respect to x is (c) The derivative of ln(x) with respect to x is (b) The derivative of 2^x with respect to x is (d) The derivative of log2(x) with respect to x is

2. Complete the following derivatives.
(a) The derivative of e^x with respect to x is
(c) The derivative of ln(x) with respect to x is
(b) The derivative of 2^x with respect to x is
(d) The derivative of log2(x) with respect to x is

Transcript text: 2. Complete the following derivatives. (a) $\frac{d}{d x}\left(e^{x}\right)=$ (c) $\frac{d}{d x} \ln (x)=$ (b) $\frac{d}{d x}\left(2^{x}\right)=$ (d) $\frac{d}{d x} \log _{2}(x)=$

Solution

Solution Steps

To solve the derivatives, we will use the rules of differentiation for exponential and logarithmic functions.

(a) The derivative of $ e^x $ with respect to $ x $ is $ e^x $.

(b) The derivative of $ 2^x $ with respect to $ x $ is $ 2^x \ln(2) $.

(d) The derivative of $ \log_2(x) $ with respect to $ x $ is $ \frac{1}{x \ln(2)} $.

Step 1: Derivative of $ e^x $

The derivative of the function $ f_1 = e^x $ is given by: \[ \frac{d}{dx}(e^x) = e^x \]

Step 2: Derivative of $ 2^x $

The derivative of the function $ f_2 = 2^x $ is calculated as follows: \[ \frac{d}{dx}(2^x) = 2^x \ln(2) \]

Step 3: Derivative of $ \ln(x) $

The derivative of the function $ f_3 = \ln(x) $ is: \[ \frac{d}{dx}(\ln(x)) = \frac{1}{x} \]

Final Answer

The derivatives are:

$ \frac{d}{dx}(e^x) = e^x $
$ \frac{d}{dx}(2^x) = 2^x \ln(2) $
$ \frac{d}{dx}(\ln(x)) = \frac{1}{x} $

Thus, the final answers are: \[ \boxed{e^x}, \quad \boxed{2^x \ln(2)}, \quad \boxed{\frac{1}{x}} \]

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