6. Derivatives of Inverse, Exponential, & Logarithmic Functions

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 dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)

The graph of y =x^{ln x} has one horizontal tangent line. Find an equation for it.

Determine whether the following statements are true and give an explanation or counterexample.

d/dx((√2)^x) = x(√2)^{x - 1}

Find the following higher-order derivatives.

dⁿ/dxⁿ (2^x)

Higher-order derivatives Find and simplify y''.

y = 2^x x

Find the derivative of the following functions.

y = x² In x

Find the derivative of the following functions.

y = In |sin x|

Find the derivative of the following functions.

y = In √x⁴+x²

Find the derivative of the following functions.

y = x In x - x

Find the derivative of the following functions.

y = In |x²-1|

Find the derivative of the following functions.

y = x² (1 - In x²)

Find the derivative of the following functions.

y = In(cos² x)

Find the derivative of the following functions.

y = In x / (In x + 1)

Find the derivative of the following functions.

y = In(e^x + e^-x)

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

y = 4 log₃(x²−1)