High School

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1.2 Use differentiation rules to determine the derivatives of the following, leaving your answer with positive exponents and in surd form where possible:

1.2.1 y = 3 sin(3x) + 2x^5 + 6 ln(x) + sin(30°)

1.2.2 y = -2e^(5x) + 4√x - 2/x

1.2.3 f(x) = √[4]{9x-8}

Answer :

To solve this problem, we will use differentiation rules to find the derivatives of the given functions. Let's tackle each part one by one.

  1. Derivative of [tex]y = 3 \sin(3x) + 2x^5 + 6 \ln(x) + \sin(30^\circ)[/tex]:

    • The derivative of [tex]3 \sin(3x)[/tex] is obtained by using the chain rule. The derivative of [tex]\sin(3x)[/tex] is [tex]3 \cos(3x)[/tex], so the derivative is:
      [tex]\frac{d}{dx}(3 \sin(3x)) = 3 \times 3 \cos(3x) = 9 \cos(3x).[/tex]

    • The derivative of [tex]2x^5[/tex] using the power rule is:
      [tex]\frac{d}{dx}(2x^5) = 10x^4.[/tex]

    • The derivative of [tex]6 \ln(x)[/tex] is:
      [tex]\frac{d}{dx}(6 \ln(x)) = \frac{6}{x}.[/tex]

    • The derivative of a constant [tex]\sin(30^\circ) = \frac{1}{2}[/tex] is zero.

    Therefore, the derivative [tex]\frac{dy}{dx}[/tex] is:
    [tex]9 \cos(3x) + 10x^4 + \frac{6}{x}.[/tex]

  2. Derivative of [tex]y = -2e^{5x} + 4\sqrt{x} - \frac{2}{x}[/tex]:

    • The derivative of [tex]-2e^{5x}[/tex] using the chain rule is:
      [tex]\frac{d}{dx}(-2e^{5x}) = -2 \times 5e^{5x} = -10e^{5x}.[/tex]

    • The derivative of [tex]4\sqrt{x}[/tex] which is [tex]4x^{1/2}[/tex] using the power rule is:
      [tex]\frac{d}{dx}(4x^{1/2}) = 4 \times \frac{1}{2} x^{-1/2} = 2x^{-1/2}.[/tex]

    • The derivative of [tex]-\frac{2}{x}[/tex] which is [tex]-2x^{-1}[/tex] using the power rule is:
      [tex]\frac{d}{dx}(-2x^{-1}) = 2x^{-2} = \frac{2}{x^2}.[/tex]

    Thus, the derivative [tex]\frac{dy}{dx}[/tex] is:
    [tex]-10e^{5x} + \frac{2}{\sqrt{x}} + \frac{2}{x^2}.[/tex]

  3. Derivative of [tex]f(x) = \sqrt[4]{9x-8}[/tex]:

    • This can be rewritten as [tex](9x - 8)^{1/4}[/tex]. Using the chain rule and the power rule:
      • The outside function is [tex]u^{1/4}[/tex], and its derivative is [tex]\frac{1}{4}u^{-3/4}[/tex].
      • The inside function [tex]u = 9x - 8[/tex] has a derivative [tex]9[/tex].

    Therefore, the derivative [tex]f'(x)[/tex] is:
    [tex]\frac{1}{4}(9x - 8)^{-3/4} \cdot 9 = \frac{9}{4}(9x - 8)^{-3/4}.[/tex]

These solutions give you the derivatives of each function, keeping the exponents positive whenever possible and using surd form where applicable.

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