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Recall the differentiation rule for a logarithm with an arbitrary base.

Chemistry

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2 parts
c

3(3x−2)ln⁡b\frac{3}{(3x-2) \ln b}

d

1sin⁡xcos⁡x\frac{1}{\sin x \cos x}

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8 steps

My apologies for the previous misunderstanding. I will now solve the problems labeled c and d directly from the image you provided.

Part c· 3 steps

Differentiate y=log⁡b(3x−2)y = \log_b (3x-2).

  1. Recall the differentiation rule for a logarithm with an arbitrary base.

    If y=log⁡b(f(x))y = \log_b(f(x)), then dydx=1f(x)ln⁡b⋅f′(x)\frac{dy}{dx} = \frac{1}{f(x) \ln b} \cdot f'(x).

  2. Identify f(x)f(x) and find its derivative.

    Here, f(x)=3x−2f(x) = 3x-2.
    The derivative of f(x)f(x) is f′(x)=ddx(3x−2)=3f'(x) = \frac{d}{dx}(3x-2) = 3.

  3. Substitute f(x)f(x) and f′(x)f'(x) into the differentiation rule.

    dydx=1(3x−2)ln⁡b⋅3\frac{dy}{dx} = \frac{1}{(3x-2) \ln b} \cdot 3 dydx=3(3x−2)ln⁡b\frac{dy}{dx} = \frac{3}{(3x-2) \ln b}

    The derivative of y=log⁡b(3x−2)y = \log_b (3x-2) is 3(3x−2)ln⁡b\boxed{\frac{3}{(3x-2) \ln b}}.

Part d

1sin⁡xcos⁡x\frac{1}{\sin x \cos x}

Differentiate y=log⁡e(tan⁡x)y = \log_e (\tan x).

  1. Rewrite the function using the natural logarithm.

    Since log⁡ex=ln⁡x\log_e x = \ln x, the function can be written as y=ln⁡(tan⁡x)y = \ln(\tan x).

  2. Recall the differentiation rule for the natural logarithm.

    If y=ln⁡(f(x))y = \ln(f(x)), then dydx=1f(x)⋅f′(x)\frac{dy}{dx} = \frac{1}{f(x)} \cdot f'(x).

  3. Identify f(x)f(x) and find its derivative.

    Here, f(x)=tan⁡xf(x) = \tan x.
    The derivative of f(x)f(x) is f′(x)=ddx(tan⁡x)=sec⁡2xf'(x) = \frac{d}{dx}(\tan x) = \sec^2 x.

  4. Substitute f(x)f(x) and f′(x)f'(x) into the differentiation rule.

    dydx=1tan⁡x⋅sec⁡2x\frac{dy}{dx} = \frac{1}{\tan x} \cdot \sec^2 x
  5. Simplify the expression using trigonometric identities.

    Recall that tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} and sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}.

    dydx=1sin⁡xcos⁡x⋅1cos⁡2x\frac{dy}{dx} = \frac{1}{\frac{\sin x}{\cos x}} \cdot \frac{1}{\cos^2 x} dydx=cos⁡xsin⁡x⋅1cos⁡2x\frac{dy}{dx} = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos^2 x} dydx=1sin⁡xcos⁡x\frac{dy}{dx} = \frac{1}{\sin x \cos x}

    This can also be written as csc⁡xsec⁡x\csc x \sec x.
    The derivative of y=log⁡e(tan⁡x)y = \log_e (\tan x) is 1sin⁡xcos⁡x\boxed{\frac{1}{\sin x \cos x}}.

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