
Math`F(x) =int_0^(e^(2x)) ln(t+1)dt` `F'(x)=?` Take note that if the function has a form `F(x) = int_a^(u(x)) f(t)dt` its derivative is `F'(x)=f(u(x))*u'(x)` Applying this formula, the derivative of...

Math`F(x)=int_pi^(lnx) cos(e^t)dt` `F'(x)=?` Take note that if the function has a form `F(x)=int_a^(u(x)) f(t)dt` its derivative is `F'(x)=f(u(x))*u'(x)` Applying this formula, the derivative of the...

MathFind the derivative if `y=e^(2x)tan(2x) ` : Let u=2x and rewrite as: `y=e^utanu ` Use the product rule noting that `d/(du)e^u=e^udu,d/(du)tanu=sec^2udu ` : `(dy)/(du)=e^udutanu+e^usec^2udu ` Since...

MathFind the derivative if `y=e^x(sinx+cosx) ` : Use the product rule: `(dy)/(dx)=e^(x)(cosxsinx)+e^(x)(sinx+cosx) ` `(dy)/(dx)=2e^xcosx `

MathFind the derivative if `y=(e^(2x))/(e^(2x)+1) ` : Use the quotient rule to get: `(dy)/(dx)=((e^(2x)+1)(2e^(2x))(e^(2x)*2e^(2x)))/(e^(2x)+1)^2 ` `(dy)/(dx)=(2e^(4x)+2e^(2x)2e^(4x))/(e^(2x)+1)^2 `...

MathFind the derivative if ` y=(e^x+1)/(e^x1) ` : Use the quotient rule to get: `(dy)/(dx)=((e^x1)(e^x)(e^x+1)(e^x))/(e^x1)^2 ` `(dy)/(dx)=(e^(2x)e^xe^(2x)e^x)/(e^x1)^2 ` `...

MathWe are asked to differentiate `y=(e^xe^(x))/2 ` : We use the fact that if f(x), g(x) are differentiable functions of x then `d/(dx)(f(x)+ g(x))=d/(dx)f(x)+d/(dx)g(x) ` and ` d/(dx)e^u=e^u...

MathFind the derivative for `y=2/(e^x+e^(x)) ` : Using the quotient rule we get: `(dy)/(dx)=(2(e^xe^(x)))/(e^x+e^(x))^2 `

MathFind the derivative of `y=ln((1+e^x)/(1e^x)) ` : Use a property of the natural logarithm to rewrite as: `y=ln(1+e^x)ln(1e^x) ` If u is a differentiable function of x, then ` d/(dx)ln(u)=(du)/u `...

MathFind the derivative if ` y=ln(1+e^(2x)) ` : If u is a differentiable function of x then ` d/(dx)lnu=(u')/u, d/(dx)e^(u)=e^u *u' ` so we get: `(dy)/(dx)=(2e^(2x))/(1+e^(2x)) `

MathFind the derivative if `y=e^(3/t^2) ` : If u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx) ` . Note that `d/(dt) 3/t^2=6/t^3 ` so we get: `(dy)/(dx)=6/t^3e^(3/t^2) `

MathFind the derivative if `y=(e^(t)+e^t))^3 ` : Use the power rule ( `d/(dx)u^n=n*u^(n1)(du)/(dx) ` where u is a differentiable function of x) to get: `(dy)/(dt)=3(e^(t)+e^t)^2(e^(t)+e^t) ` If...

MathFind the derivative if ` y=x^2e^(x)` : Use the product rule to get: `(dy)/(dx)=x^2(e^(x))+2xe^(x) ` `(dy)/(dx)=e^(x)(2xx^2) `

MathFind the derivative if `y=x^3e^x ` : Use the product rule to get: `(dy)/(dx)=x^3e^x+3x^2e^x ` `(dy)/(dx)=e^x(x^3+3x^2) `

MathFind the derivative if `y=xe^(4x) ` : If u is a differentiable function of x then ` d/(dx)e^u=e^u (du)/(dx) ` , so using the product rule we get: `(dy)/(dx)=x(4)e^(4x)+e^(4x) `...

MathFind the derivative if ` y=e^xlnx ` : If u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx), d/(dx)lnu=((du)/(dx))/u ` , so using the product rule we get: `(dy)/(dx)=e^xlnx+e^x*1/x...

MathFind the derivative if `y=5e^(x^2+5) ` : Note that if u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx) ` , so using the constant rule we get: ` (dy)/(dx)=5(2x)e^(x^2+5) `...

MathFind the derivative if ` y=e^(x4) ` : If u is a differentiable function of x, then `d/(dx)e^u=e^u (du)/(dx) ` . Here u=x4 and du=1 so: `(dy)/(dx)=e^(x4) `

Math`y=e^(2x^3)` Find the derivative using the chain rule. `y'=e^(2x^3)*(6x^2)` `y'=6x^2e^(2x^3)` The answer is `6x^2e^(2x^3).`

Math`y=e^sqrtx` `y'=e^sqrt(x)*1/2x^(1/2)` `y'=e^sqrt(x)/[2sqrt(x)]` `y'=[sqrt(x)e^sqrt(x)]/[2x]` The derivative is `[sqrt(x)e^sqrt(x)]/[2x]`

Math`y=e^(8x)` `y'=e^(8x)*(8)` `y'=8e^(8x)` The answer is `8e^(8x)`

Math`f(x)=e^(2x)` Find the derivative using the Chain Rule. `f'(x)=e^(2x)*2` `f'(x)=2e^(2x)` The answer is `2e^(2x)`

MathWe are asked to determine whether the function ` y=ln(sqrt(x+2))` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative....

MathWe are asked to show that the function `y=ln(4x) ` has an inverse by showing, using the derivative, that the function is monotonic on its entire domain. The domain for the function is x>0....

MathWe are asked to determine if the function `y=ln(x3) ` has an inverse function by determining if the function is strictly monotonic on its entire domain using the derivative. `y'=1/(x3) ` The...

MathWe are asked to determine if ` ln(x^2) ` has an inverse; there is an inverse if the function is monotonic on its entire domain, so we use the derivative to determine the monotonicity:...

MathWe are asked to use the derivative to determine if ` y=lnx ` is monotonic on its domain. The domain of the natural logarithm is x>0: `y'=1/x ` and `1/x > 0 forall x>0 ` so the function is...

MathWe are asked to use the derivative of ` y=5000/(1+e^(2x)) ` to determine if the function has an inverse. (We can show that an inverse exists if the function is monotonic on its entire domain.)...

MathWe are asked to determine if the function ` y=e^x ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all reals....

MathWe are asked to determine if the function `y=e^(ln(3x)) ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain of the...

Math`e^(lnx)=4` To solve, apply the logarithm rule `e^(lnm) = m` . So the left side simplifies to: `x=4` Therefore, the solution of the given equation is x=4.

MathWe are asked to show that `f(x)=x+2,[2,oo) ` has an inverse by showing that the function is monotonic on the interval using the derivative: By definition, ` f(x)=x+2={[x+2,x+2...

MathGiven `f(x)=(x4)^2, [4,oo) ` we are asked to show that the function has an inverse since it is monotonic on the given interval: `f'(x)=2(x4) ` and `2(x4)>=0 forall x>=4 ` so the function...

MathWe are asked to determine if `y=x^4/42x^2 ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain for a polynomial is all...

Math`f(x)=2xx^2` `f'(x)=12x=0` `12x=0` `2x=1` `x=1/2` A critical value for x is `x=1/2.` In the interval `(oo, 1/2)` the function increases. In the interval `(1/2,oo)` the function...

Math360 square tiles of length 30cm were needed to cover the floor of a room which is 7 m 20 cm long....We are given that 360 tiles were required to tile a floor: the tiles are square and 30 cm on edge. The length of the room is 7m 20cm, and we are asked to determine the width of the room assuming...

MathHello! Any (straight) line perpendicular to the xaxis has the equation of the form `x = a,` where `a` is a constant. In other words, for all points of such a line the xcoordinate is the same (and...

MathHello! I suppose that "sinx/2+cosx1=0" means "sin(x/2)+cos(x)1=0" (it can also mean "(sinx)/2+cos(x)1=0" and "sin(x/2)+cos(x1)=0"). To solve this equation, recall the double angle formula...

MathFirst, draw the triangle formed by the three equations x+y=1, x=1 and y=1. Let the vertices of the triangle be A, B and C (see attached figure). Base on the graph, the coordinates of the vertices...

MathA tourist in France wants to visit 5 different cities. If the route is randomly selected, what is...In order to answer this question, let's take a look at the reasoning behind it! There are many different kinds of probability problems, but this one relates to the Factorial Rule, which is: n! = n...

Math`2+secxcscx=(sinx+cosx)^2/(sinxcosx)` To prove, consider the left side of the equation. `2+secxcscx` Express the secant and cosecant in terms of cosine and sine, respectively. `=2+1/cosx*1/sinx`...

MathHello! This figure is really a parallelogram, for example because the opposite sides have the same length: `AB = CD = 2sqrt(2)` and `BC = AD = 2sqrt(37).` Or we can check that the opposite...

MathHello! The most clear method to solve this is to denote the variables and to solve the system of equations that occurs. This system is simple! Let `S` be the amount of dollars Soumik have, and `K`...

MathHello! Denote the slope of this line as `m.` The vertical line (which has an undefined slope) doesn't suit us, so we'll not miss a solution. Horizontal line with `m=0` doesn't suit also, so we can...

MathWe are given a+b+c=395, b+c+d=1001 and a<b<c<d and we are asked to find the value of d: As stated, d can take on a range of values. Subtract the two equations: b+c+d=1001a+b+c...

MathIn order to find the greatest possible value of the highest number, we must consider the lowest possible values of five of the numbers. We are given the information that all the numbers are...

MathGiven ` x^15x^13+x^11x^9+x^7x^5+x^3x=7 ` , we are asked to show that `x^16>15 ` : First, note that `x^15x^13+x^11x^9+x^7x^5+x^3x=x(x1)(x+1)(x^4+1)(x^8+1) ` so the polynomial has real...

MathHello! As I understand, the deceleration is uniform (the same all the time). Denote it as `agt0` and denote the initial speed as `V_0.` In m/s `V_0 = 140/3.6.` Then the speed is `V(t) = V_0  a*t`...

MathDenote the numbers as `a_1 lt= a_2 lt= a_3 lt= a_4 lt= a_5 lt= a_6 lt= a_7.` It is given that: `(a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)/7 = 12,` `(a_1 + a_2 + a_3 + a_4)/4 = 8,` `(a_4 + a_5 +...

MathHere are some areas of interest and questions that arise in mathematics teaching in the elementary grades: Curriculum  What content should be taught? When should it be taught? What are the...
rows
eNotes
search