How to solve a classmate draws a parallelogramfor whichone side is twice as long as the other. If one side is 26 units, what are
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The derivatives of the functions are listed below:
(a)
(b)
(c) f'(x) = [(cos x + sin x) · (x² - 1) - (sin x - cos x) · (2 · x)] / (x² - 1)²
(d) f'(x) = (5ˣ · ㏑ 5) · ㏒₅ x + 5ˣ · [1 / (x · ㏑ 5)]
(e) f'(x) = 45 · (x⁻⁵ + √3)⁻⁸ · x⁻⁶
(f)
(g)
(h) f'(x) = cot x + cos (㏑ x) · (1 / x)
<h3>How to find the first derivative of a group of functions</h3>
In this question we must obtain the <em>first</em> derivatives of each expression by applying <em>differentiation</em> rules:
(a)
Given
Definition of power
Derivative of constant and power functions / Derivative of an addition of functions / Result
(b)
Given
Definition of power
Derivative of a product of functions / Derivative of power function / Rule of chain / Result
(c) f(x) = (sin x - cos x) / (x² - 1)
- f(x) = (sin x - cos x) / (x² - 1) Given
- f'(x) = [(cos x + sin x) · (x² - 1) - (sin x - cos x) · (2 · x)] / (x² - 1)² Derivative of cosine / Derivative of sine / Derivative of power function / Derivative of a constant / Derivative of a division of functions / Result
(d) f(x) = 5ˣ · ㏒₅ x
- f(x) = 5ˣ · ㏒₅ x Given
- f'(x) = (5ˣ · ㏑ 5) · ㏒₅ x + 5ˣ · [1 / (x · ㏑ 5)] Derivative of an exponential function / Derivative of a logarithmic function / Derivative of a product of functions / Result
(e) f(x) = (x⁻⁵ + √3)⁻⁹
- f(x) = (x⁻⁵ + √3)⁻⁹ Given
- f'(x) = - 9 · (x⁻⁵ + √3)⁻⁸ · (- 5) · x⁻⁶ Rule of chain / Derivative of sum of functions / Derivative of power function / Derivative of constant function
- f'(x) = 45 · (x⁻⁵ + √3)⁻⁸ · x⁻⁶ Associative and commutative properties / Definition of multiplication / Result
(f)
Given
Rule of chain / Derivative of sum of functions / Derivative of multiplication of functions / Derivative of logarithmic functions / Derivative of potential functions
Distributive property / Result
(g)
Given
Derivative of the subtraction of functions / Derivative of arccosine / Derivative of arctangent / Rule of chain / Derivative of power functions / Result
(h) f(x) = ㏑ (sin x) + sin (㏑ x)
- f(x) = ㏑ (sin x) + sin (㏑ x) Given
- f'(x) = (1 / sin x) · cos x + cos (㏑ x) · (1 / x) Rule of chain / Derivative of sine / Derivative of natural logarithm /Derivative of addition of functions
- f'(x) = cot x + cos (㏑ x) · (1 / x) cot x = cos x / sin x / Result
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