Try It Yourself#

Exercise 1#

Find all values of \(t\) such that

\[25^{(t^2-2t-1)} = \frac{1}{5^{t}}.\]
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Answer: \(t = 2\), \(t=-1/2\)

Exercise 2#

Find all values of \(x\) such that

\[3^{2x}-30\cdot 3^x +81=0.\]
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Answer: \(x = 3\), \(x=1\)

Exercise 3#

Expand and simplify

\[\ln \left(\dfrac{x}{e^{3x}}\right) - \ln(x^25^x).\]
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Answer: \(-3x-\ln(x) - x\ln(5)\)

Exercise 4#

Compute the derivative of

\[f(t) = \displaystyle e^{5t^3+3}.\]
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Answer: \(15t^2e^{5t^3+3}\)

Exercise 5#

Compute the derivative of

\[f(t) = \displaystyle \ln(5t^3+3).\]
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Answer: \(15t^2/(5t^3+3)\)

Exercise 6#

Compute the derivative of

\[\ln\left(\dfrac{4x^3}{2x-1}\right).\]
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Answer: \(\frac{3}{x} - \frac{2}{2x-1}\)

Exercise 7#

Given the supply equation \(s(x)\) for video accelerator boards to be

\[p=x \ln(x),\]

determine the marginal revenue function, \(R'(x)\).

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Answer: \(x(1+2\ln(x))\)

Exercise 8#

Find the equation of the tangent line to the curve

\[\displaystyle y=\frac{\ln (x^2)}{x^3}\]

at the point \((1,0)\).

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Answer: \(y=2(x-1)\)

Exercise 9#

Let

\[\displaystyle e^{xy} -x^6 = y^5.\]

Find \(\displaystyle \frac{dy}{dx}\).

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Answer: \(y' = (6x^5 - ye^{xy})/(xe^{xy}-5y^4)\)

Exercise 10#

Let

\[\displaystyle \ln(3x - 2y) + y^4 = 1.\]

Find \(\displaystyle \frac{dy}{dx}\) at the point \((1,1)\).

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Answer: \(y' = -3/2\)

Exercise 11#

Use logarithmic differentiation to compute the derivative of

\[\dfrac{x^5 (x^3+4x)^7}{\sqrt{6x+2}}.\]
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Answer: \(\frac{x^5 (x^3+4x)^7}{\sqrt{6x+2}}\left[\frac{5}{x} + \frac{7(3x^2 + 4)}{x^3+4x} - \frac{3}{6x+2}\right]\)

Exercise 12#

Compute the derivative of

\[(x^2 + 3)^{7x+1}.\]
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Answer: \((x^2 + 3)^{7x+1}\left[7\ln(x^2+3) + \frac{2x(7x+1)}{x^2+3}\right]\)