Question

derivatives of transcendentals
4 sinusoidals, exponentials, logs 1, 1, 4, 5

1. find the first and second derivatives of the following functions with respect to x

a) $e^x$
b) $e^{2x}$
c) $e^{-x}$
d) $e^{-5x}$
e) $cos x$
f) $sin(-x)$
g) $-sin(2x)$
h) $-cos(pi/2)$
i) $ln x$
j) $ln sqrt{x}$
k) $ln(8x)$
l) $ln(xe^x)$
determine the domain where the functions are concave up or down. for the derivatives of the logarithms, using some log rules for powers and products may help.

Explanation:

Step1: Find first derivative of \( e^x \)

The derivative of \( e^x \) with respect to \( x \) is \( e^x \) (by the rule \( \frac{d}{dx}e^x = e^x \)).
\( f'(x) = e^x \)

Step2: Find second derivative of \( e^x \)

Differentiate \( f'(x) = e^x \) again. Using the same rule, \( \frac{d}{dx}e^x = e^x \), so \( f''(x) = e^x \).

(We can solve other parts similarly. For example, for \( e^{2x} \):

Step1: First derivative of \( e^{2x} \)

Using the chain rule \( \frac{d}{dx}e^{u} = e^u \cdot u' \), where \( u = 2x \), \( u' = 2 \). So \( f'(x) = e^{2x} \cdot 2 = 2e^{2x} \).

Step2: Second derivative of \( e^{2x} \)

Differentiate \( 2e^{2x} \). Again using chain rule, \( u = 2x \), \( u' = 2 \), so \( f''(x) = 2 \cdot e^{2x} \cdot 2 = 4e^{2x} \).

For \( \cos x \):

Step1: First derivative of \( \cos x \)

The derivative of \( \cos x \) is \( -\sin x \), so \( f'(x) = -\sin x \).

Step2: Second derivative of \( \cos x \)

Differentiate \( -\sin x \), the derivative of \( \sin x \) is \( \cos x \), so \( f''(x) = -\cos x \).

And so on for other functions.)

Answer:

For \( \boldsymbol{e^x} \): First derivative \( \boldsymbol{e^x} \), Second derivative \( \boldsymbol{e^x} \)
For \( \boldsymbol{e^{2x}} \): First derivative \( \boldsymbol{2e^{2x}} \), Second derivative \( \boldsymbol{4e^{2x}} \)
For \( \boldsymbol{\cos x} \): First derivative \( \boldsymbol{-\sin x} \), Second derivative \( \boldsymbol{-\cos x} \)
(And similar solutions for other functions can be derived using derivative rules like chain rule, product rule, and basic derivative formulas for trigonometric, exponential, and logarithmic functions.)