Reminder: in these notes, we use \(\log x\) to mean \(\log_e x\text{,}\) which is also commonly written elsewhere as \(\ln x\text{.}\)
The volume in decibels (dB) of a sound is given by the formula:
where \(P\) is the intensity of the sound and \(S\) is the intensity of a standard baseline sound. (That is: \(S\) is some constant.)
How much noise will ten speakers make, if each speaker produces 3dB of noise? What about one hundred speakers?
An investment of $1000 with an interest rate of 5% per year grows to
dollars after \(t\) years. When will the investment double?
Which of the following expressions, if any, is equivalent to \(\log\left(\cos^2 x\right)\text{?}\)
Differentiate \(f(x)=\log(10x)\text{.}\)
Differentiate \(f(x)=\log(x^2)\text{.}\)
Differentiate \(f(x)=\log(x^2+x)\text{.}\)
Differentiate \(f(x)=\log_{10}x\text{.}\)
Find the derivative of \(y=\dfrac{\log x}{x^3}\text{.}\)
Evaluate \(\ds\diff{}{\theta} \log(\sec \theta)\text{.}\)
Differentiate the function \(f(x)=e^{\cos\left(\log x\right)}\text{.}\)
Evaluate the derivative. You do not need to simplify your answer.
Differentiate \(\sqrt{-\log(\cos x)}\text{.}\)
Calculate and simplify the derivative of \(\log\big(x+\sqrt{x^2+4}\big)\text{.}\)
Evaluate the derivative of \(g(x)=\log (e^{x^2}+\sqrt{1+x^4})\text{.}\)
Evaluate the derivative of the following function at \(x=1\text{:}\) \(g(x)=\log\Big(\dfrac{2x-1}{2x+1}\Big)\text{.}\)
Evaluate the derivative of the function \(f(x) = \log\left(\sqrt{\dfrac{(x^2+5)^3}{x^4+10}}\right)\text{.}\)
Evaluate \(f'(2)\) if \(f(x) = \log\big(g\big(xh(x)\big)\big)\text{,}\) \(h(2) = 2\text{,}\) \(h'(2) = 3\text{,}\) \(g(4) = 3\text{,}\) \(g'(4) = 5\text{.}\)
Differentiate the function
Differentiate \(f(x)=x^x\text{.}\)
Find \(f'(x)\) if \(f(x) = x^x+\log_{10}x\text{.}\)
Differentiate \(f(x) = \sqrt[4]{\dfrac{(x^4+12)(x^4-x^2+2)}{x^3}}\text{.}\)
Differentiate \(f(x)=(x+1)(x^2+1)^2(x^3+1)^3(x^4+1)^4(x^5+1)^5\text{.}\)
Differentiate \(f(x) = \left(\dfrac{5x^2+10x+15}{3x^4+4x^3+5}\right)\left(\dfrac{1}{10(x+1)}\right)\text{.}\)
Let \(f(x) = (\cos x)^{\sin x}\text{,}\) with domain \(0 \lt x \lt \tfrac{\pi}{2}\text{.}\) Find \(f'(x)\text{.}\)
Find the derivative of \((\tan(x))^x\text{,}\) when \(x\) is in the interval \((0,\pi/2)\text{.}\)
Find \(f'(x)\) if \(f(x)= (x^2+1)^{(x^2+1)}\)
Differentiate \(f(x)= (x^2+1)^{\sin(x)}\text{.}\)
Let \(f(x)= x^{\cos^3(x)}\text{,}\) with domain \((0,\infty)\text{.}\) Find \(f'(x)\text{.}\)
Differentiate \(f(x)= (3+\sin(x))^{x^2-3}\text{.}\)
Let \(f(x)\) and \(g(x)\) be differentiable functions, with \(f(x) \gt 0\text{.}\) Evaluate \(\ds\diff{}{x}\left\{[f(x)]^{g(x)}\right\}\text{.}\)
Let \(f(x)\) be a function whose range includes only positive numbers. Show that the curves \(y=f(x)\) and \(y=\log(f(x))\) have horizontal tangent lines at the same values of \(x\text{.}\)