
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 if the function `y=ln(x2)^2 ` has an inverse by finding out if the function is strictly monotonic on its domain by using the derivative. The domain is `RR{2} ` . `...

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=800/(100e^(2/x)) ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is `x...

MathWe are asked to determine if the function `y=100e^(2x) ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real...

MathWe are asked to determine if the function ` y=50e^(x)` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real...

MathWe are asked to determine if the function ` y=8e^x12 ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real...

MathWe are asked to determine if the function `y=92e^x ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real...

MathWe are asked to determine if the function `y=5e^x ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real...

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.

Math`f(x)=sec(x)` Take note that a function is strictly monotonic on a given interval if it is entirely increasing on that interval or entirely decreasing on that interval. To determine if f(x) is...

MathWe are asked to verify that the function f(x)=cos(x) defined on the interval `[0,pi) ` has an inverse function by determining that the function is strictly monotonic on the interval using the...

MathWe are asked to determine if the function `y=cot(x) "on" (0,pi) ` has an inverse function by finding if the function is strictly monotonic on the interval using the derivative. `y'=csc^2(x) ` . On...

Math`f(x)=4/x^2` `f(x)=4x^(2)` `f'(x)=8x^(3)=0` There are no critical values for x in the domain (0, `oo).` If you choose any x value in the domain `(0,oo),` `f'(x)=8x3<0` therefore the...

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 the function `y=cos((3x)/2) ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real...

Math`f(x)=ln(x3)` Take note that a function is strictly monotonic if it is increasing on its entire domain or decreasing on its entire domain. For our function, `f(x)=ln(x3)` to determine if it is...

MathWe are asked to determine if the function `y=x^5+2x^3 ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain for...

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)=x^36x^2+12x` `f'(x)=3x^212x+12=0` `3(x^24x+4)=0` `3(x2)(x2)=0` `x=2` A critical x value is `x=2.` The function increases in the interval `(oo, 2).` The function increases in the...

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...
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