58–59. Carry out the following steps.

b. Find the slope of the curve at the given point.

xy^5/2+x^3/2y=12; (4, 1)

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x²(3y²−2y³) = 4

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x(1−y²)+y³=0

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

c. d/dx ((f(x)g(x)) |x=3

Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).

Find d/dx(ln√x²+1).

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

y = e^x x^e

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

y = In (x³+1)^π

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

y = 5^3t

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

y = 4^-x sin x

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

y = 10^In 2x

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

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

g (x) = x^ In x; a = e

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

h (x) = x^√x; a = 4

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

f (x) = (sin x)^In x; a = π/2

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

f (x) = (4 sin x+2)^cos x; a = π

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.

Find the following higher-order derivatives.

dⁿ/dxⁿ (2^x)

State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

Explain why b^x = e^xlnb.

Simplify the expression e^xln(x²+1).

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