3.4: Derivatives of Exponential and Logarithmic Functions

x ln x + x ln x =? For AP question given a constant k and • x, the derivative will be the? Given sin^-6 (x) - cos^3 (x): use? no need to use? For sin (9pie-1), make sure to add? To facilitate product rule, use brackets around a?

2x lnx,constant, chain rule, quotient rule, parentheses, larger equation,

logarithmic differentiation: take the ln of? then use the? Use product rule for d/dx (x • ln7). Move y to the? plug in the? for x^ (4- root 5), use the regular? ex. r • x^r-1. for ln [10x]^ln10x, pull the? create a?then find the?

both sides, base rule, right, original equation, power rule, ln10x down, squared parentheses, derivative

Word problems with y = P(t): to find the rate of change, use the? then plug in? To find the max rate or change: graph the? use the? e limit equation? given 5^log 5 (8x) =? log and the base? e^ln a^x =? e and ln? use which rules to find the derivative? if the function is on the bottom, it likely had a?

chain rule, x values, derivative, trace feature, e = limit n approaches infinity (1 + 1/n)^n, 8x, cancel out, a^x, cancel out, power chain rule, negative parentheses exponent

Ln2 is a? use constant? To find where the tangent line at a graph is perpendicular to another libe: find derivative of? make them? take the - reciprocal of the side with a? solve for? plug into original equation to find?

constant, derivative rule, both equations, equal to each other, whole #, x, y,

Derivative of e^x? ex. dy/dx of e^ (x +x^2)? derivative of a^x? For a problem like 2t • ln2 = 2t: take the natural log ln of both? remember: three log rules? (3) ex. ln 2t = t • ln2 for power rule.

d/dx e^u = e^x • du/dx, e^ (x + x^2) • 2x +1, d/dx a^u = a^u • ln a • du/dx, sides, product rule/ quotient rule/ power rule,

derivative of log a u? power rule for arbitrary powers? ex. x^ square root of 2 = y ? to find domain of f' (c): look at the? estimate the? for logarithmic differentiation, use implicit differentiation?

d/dx log a u = 1/(u ln a) • du/dx, d/dx u^n = n • u^n-1 • du/dx, square root of 2 • x^square root of 2 -1 • 1, equation, domain, rules,

To find a point on which the tangent line is parallel to another line: first, find the derivatives of both? make them? solve for? Then plug x into the specified graph where the? write an? log Quotient rule? product rule? power rule? given x^pie: use which equation to solve? not the?

equations, equal to each other, x, point is, ordered pair (x,y), log x-log a= log x/log a, log xa = logx + loga, log x^a = a • log x, regular power rule, chain power rule,

the domain of f' (x) is the domain of? find the domain of the? (for the?) write 4cos4x/sin4x as? for the online platform, always add parentheses around the? ex. ln (4x). given a huge equation for logarithmic differentiation; pit the radical as an exponent of the? use the quotient log rule to separate the? find the derivative of each? this rule is the?

f, original function for the domain of the derivative, 4cot4x, inner x value,parentheses, huge fraction, ln function, sum and difference rule,

for ln (2t / ln2), write? derivative of ln x? for tangent slope through the origin equation: find two points and plug them into? make them equal to? then? use the different?

ln2t - ln (ln2), d/dx lnu = 1/u • du/dx, y2-y1/x2-x1, m value, solve, exponent rules,

given (5x)^ 4, remember to use? this is not a? a base function a^x has a is a? ex. 5^x. The function is in the? not the? To use logarithmic differentiation: take the ln of? remember that for every y derivative, write? If you have y on the left side and dy/dx on the right, repalce y with the? ex. y ln 6x and y = 5x-1 becomes (5x-1) (ln6x)

regular power rule, base or log function, whole number, exponent, base, both sides, dy/dx, original equation y = function

to find value of m from y = ln2x: find? plug derivative into the equation? Solve for? then plug the x value into the? domain of ln (2-cosx) is? for ln • x^tanx, bring down? exact value of lim h approaches 0 2^h-1/h is? Remembe inverse rule? use? rearrange -cost = sin t to make?

y', y = mx, x, derivative equation, all real #'s, tanx, ln2, f (f^-1 (x) ) = x, implicit differentiation, tan t,