Try It Yourself#
Exercise 1#
Solve for unknown value in the exponent
Find all values of \(t\) such that \(25^{t^2-2t-1} = \dfrac{1}{5^{t}}\).
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Answer: \(t = 2\), \(t=-1/2\)
Exercise 2#
Solve for unknown value in the exponent
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#
Simplify
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
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
Compute the derivative of \(f(t) = \displaystyle \ln(5t^3+3)\).
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Answer: \(\dfrac{15t^2}{5t^3+3}\)
Exercise 6#
Compute the derivative
Compute the derivative of \(\ln\left(\dfrac{4x^3}{2x-1}\right)\).
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Answer: \(\dfrac{3}{x} - \dfrac{2}{2x-1}\)
Exercise 7#
Marginal revenue function
Given the supply equation \(s(x)\) for video accelerator boards to be
determine the marginal revenue function, \(R'(x)\).
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Answer: \(x(1+2\ln(x))\)
Exercise 8#
Equation of a tangent line
Find the equation of the tangent line to the curve \(\displaystyle y=\frac{\ln (x^2)}{x^3}\) at \(x=1\).
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Answer: \(y=2(x-1)\)
Exercise 9#
Implicit differentiation
Let \(\displaystyle e^{xy} -x^6 = y^5\). Compute \(\displaystyle \frac{dy}{dx}\).
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Answer: \(y' = \dfrac{6x^5 - ye^{xy}}{xe^{xy}-5y^4}\)
Exercise 10#
Implicit differentiation
Let \(\displaystyle \ln(3x - 2y) + y^4 = 1\). Compute \(\displaystyle \frac{dy}{dx}\) at the point \((1,1)\).
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Answer: \(y' = -3/2\)
Exercise 11#
Logarithmic differentiation
Use logarithmic differentiation to compute the derivative of \(\dfrac{x^5 (x^3+4x)^7}{\sqrt{6x+2}}\).
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Answer: \(\dfrac{x^5 (x^3+4x)^7}{\sqrt{6x+2}}\left[\dfrac{5}{x} + \dfrac{7(3x^2 + 4)}{x^3+4x} - \dfrac{3}{6x+2}\right]\)
Exercise 12#
Logarithmic differentiation
Compute the derivative of \((x^2 + 3)^{7x+1}\).
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Answer: \((x^2 + 3)^{7x+1}\left[7\ln(x^2+3) + \dfrac{2x(7x+1)}{x^2+3}\right]\)