6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Find the derivative of the given function.

$g\left(t\right)=\log_5\left(7^{t^2+4}\right)$

Find the derivative of the given function.

$h\left(x\right)=\sin\left(\log_4\left(x^2\right)\right)$

Find the derivative of the given function.

$f(x)=2x^3-1+\ln x$

Find the derivative of the given function.

$g(x)=e^{x}+\ln x^5$

Find the derivative of the given function.

$h\left(x\right)=\ln\left(\frac{\sqrt{x+1}}{x^2+3}\right)$

Find the derivative of the given function.

$y=x^2\ln(x^2)$

Find the derivative of the given function.

$g(x)=e^{x^2\ln x}$

Identify the open intervals on which the function is increasing or decreasing.

$f(x)=xe^{-2x}$

Identify the local minimum and maximum values of the given function, if any.

$f(t)=t^2\ln t$, $t>0$

Determine the intervals for which the function is concave up or concave down. State the inflection points.

$f(x)=4\ln(3x^2)$

15–48. Derivatives Find the derivative of the following functions.

P = 40/1+2^-t

15–48. Derivatives Find the derivative of the following functions.

y = 10^x(In 10^x-1)

15–48. Derivatives Find the derivative of the following functions.

f(x) = 2^x/2^x+1

15–48. Derivatives Find the derivative of the following functions.

s(t) = cos 2^t

The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)

Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.