Don’t forget to use the product rule.

[latex]h^{\prime}(x)=e^{2x}+2xe^{2x}[/latex]

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[latex]
\begin{array}{rcl}
f'(x) &=& e^{\tan(2x)} \cdot \frac{d}{dx}(\tan(2x)) \\
&=& e^{\tan(2x)} \cdot \sec^2(2x) \cdot 2 \\
&=& 2e^{\tan(2x)} \sec^2(2x)
\end{array}
[/latex]

Find [latex]A^{\prime}(4)[/latex].

[latex]996[/latex]

Use a property of logarithms to simplify before taking the derivative.

[latex]f^{\prime}(x)=\frac{15}{3x+2}[/latex]

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[latex]f(x) = \ln(x^2) + \ln(\sin x) - \ln(2x+1)[/latex]

Differentiate term by term:

[latex]
\begin{array}{rcl}
f'(x) &=& \frac{2}{x} + \frac{\cos x}{\sin x} - \frac{2}{2x+1} \\
&=& \frac{2}{x} + \cot x - \frac{2}{2x+1}
\end{array}
[/latex]

Use the general logarithm derivative formula:

[latex]\frac{dy}{dx} = \frac{3}{(3x+1)\ln 2}[/latex]

Evaluate at [latex]x = 1[/latex]:

[latex]\frac{dy}{dx}\Big|_{x=1} = \frac{3}{4\ln 2} = \frac{3}{\ln 16}[/latex]

The slope of the tangent line at [latex]x = 1[/latex] is [latex]\frac{3}{\ln 16}[/latex].

Evaluate the derivative at [latex]x=2[/latex].

[latex]9 \ln (3)[/latex]

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[latex]\frac{dy}{dx}=x^x(1+\ln x)[/latex]

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[latex]y^{\prime}=\pi (\tan x)^{\pi -1} \sec^2 x[/latex]

1. Take natural log:
   [latex]\ln y = \ln(2x^4 + 1)^{\tan x}[/latex]
2. Apply log power rule:
   [latex]\ln y = \tan x \ln(2x^4 + 1)[/latex]
3. Differentiate both sides:
   [latex]\frac{1}{y}\frac{dy}{dx} = \sec^2 x \ln(2x^4 + 1) + \tan x \cdot \dfrac{8x^3}{2x^4 + 1}[/latex]
4. Solve for [latex]\frac{dy}{dx}[/latex]:
   [latex]\frac{dy}{dx} = y(\sec^2 x \ln(2x^4 + 1) + \tan x \cdot \dfrac{8x^3}{2x^4 + 1})[/latex]
5. Substitute [latex]y[/latex]:
   [latex]\frac{dy}{dx} = (2x^4 + 1)^{\tan x}(\sec^2 x \ln(2x^4 + 1) + \tan x \cdot \dfrac{8x^3}{2x^4 + 1})[/latex]

1. Take natural log:
   [latex]\ln y = \ln x^r[/latex]
2. Apply log power rule:
   [latex]\ln y = r \ln x[/latex]
3. Differentiate:
   [latex]\frac{1}{y}\frac{dy}{dx} = r \cdot \frac{1}{x}[/latex]
4. Solve for [latex]\frac{dy}{dx}[/latex]:
    [latex]\frac{dy}{dx} = y \cdot \frac{r}{x}[/latex]
5. Substitute [latex]y[/latex] and simplify:
   [latex]\frac{dy}{dx} = rx^{r-1}[/latex]