eNotes Homework Help is a way for educators to help students understand their school work. Our experts are here to answer your toughest academic questions! Once it’s posted to our site, your question could help thousands of other students.
Popular Titles
Showing
in
Math

MathTo find the derivative of the function `f(x)=10x` using the limit process use the fact that the slope is `m=lim_(h>0)(f(x+h)f(x))/(h)` `=lim_(h>0)(10(x+h)(10x))/(h)`...

MathBy limit process, the derivative of a function f(x) is : `f'(x) = lim_(h > 0) [{f(x+h)  f(x)}/h]` Now, the given function is : `f(x) = 7x 3` THus, `f'(x) =lim_(h > 0) [{f(x+h) ...

MathThe domain of a function y = f(x) is the set of all real values that x an take on for which y is real. If the value of x lies in the domain it is possible to plot the point (x, f(x)). For the...

MathThe function f(x) = x^3. To determine f(a+h) substitute x with a+h. f(a+h) = (a+h)^3 = a^3 +3*a^2*h +3*a*h^2+ h^3 f(a) = a^3 `(f(a+h)  f(a))/h` = `(a^3+3*a^2*h +3*a*h^2+ h^3  a^3)/h` = `(3*a^2*h...

MathGiven f(x)=4+3xx^2, then f(x+h)=4+3(x+h)(x+h)^2. Consider the difference f(x+h)f(x): (x+h)^2+x^2 + 3x+3h3x + 44 = x^22xhh^2+x^2 + 3h = (32x)*h  h^2. This divided by h is equal to...

MathThe vertical asymptotes of a curve are lines that the graph of the curve approaches but does not touch. For `y = (f(x))/(g(x))` , the vertical asymptotes are lines x = a where a is the root of the...

Math`d/(dt) sin^1t=1/sqrt(1t^2)` `y'=1/(sqrt(1(2x+1)^2))*d/(dx) (2x+1)` `y'=2/sqrt(1(4x^2+1+4x))` `y'=2/sqrt(14x^214x)` `y'=2/sqrt(4x^24x)` `y'=2/sqrt(4(x^2x))` `y'=2/(2(sqrt(x(x+1))))`...

MathFind `g'(x) ` if `g(x)=sqrt(x^21)sec^(1)(x) ` : Use the product rule: `g'(x)=sqrt(x^21)*1/(xsqrt(x^21))+(1/2)(x^21)^(1/2)(2x)sec^(1)(x) ` `=1/x+(xsec^(1)(x))/sqrt(x^21) ` or...

Math`d/(dt) cos^1(t)=(1)/sqrt(1t^2)` `G(x)=sqrt(1x^2)cos^1(x)` `G'(x)=sqrt(1x^2) d/(dx) cos^1(x) + cos^1(x) d/(dx)sqrt(1x^2)` `G'(x)=sqrt(1x^2)*(1/sqrt(1x^2)) + cos^1(x)...

Math`y=tan^1(xsqrt(1+x^2))` `d/(dt) tan^1t=1/(1+t^2)` `y'=1/(1+(xsqrt(1+x^2))^2) * d/(dx)(xsqrt(1+x^2))` `y'=1/(1+x^2+1+x^22xsqrt(1+x^2)) *(1(1/2)(1+x^2)^(1/2)(2x))`...

Math`d/(dx) cot^1x=(1)/(1+x^2)` `h(t)=cot^1(t) + cot^1(1/t)` `h'(t)=(1)/(1+t^2) + (1)/(1+(1/t)^2) *d/(dt) (1/t)` `h'(t)=(1)/(1+t^2) + (1)/(1+(1/t)^2) *(1t^2)` `h'(t)=(1)/(1+t^2)...

Math`d/(dx)sin^1(x)=1/sqrt(1x^2)` `F(theta)=sin^1(sqrt(sin(theta)))` `F'(theta)=(1/sqrt(1sintheta)) * d/(d theta)sqrt(sintheta)` `F'(theta)=(1/sqrt(1sintheta)) *(1/2)(sintheta)^(1/2) costheta`...

Math`y=xsin^1(x) +sqrt(1x^2)` `y'=xd/(dx) sin^1(x) + sin^1(x) d/(dx) x + d/(dx)sqrt(1x^2)` `y'=x(1/sqrt(1x^2)) + sin^1(x) + (1/2)(1x^2)^(1/2)(2x)` `y'=x/sqrt(1x^2) + sin^1(x)...

Math`d/(dx)cos^1(x)=1/sqrt(1x^2)` using above `y=cos^1(sin^1(t))` `y'=(1)/sqrt(1(sin^1(t))^2) * d/(dt) sin^1(t)` `y'=(1)/sqrt(1(sin^1(t))^2) * (1/sqrt(1t^2))`...

Math`y=cos^1((b+acos(x))/(a+bcos(x)))` `y'=((1)/sqrt(1((b+acos(x))/(a+bcos(x)))^2))* d/(dx) ((b+acos(x))/(a+bcos(x)))` `y'=(a+bcos(x))/sqrt((a+bcos(x))^2(b+acos(x))^2) *d/(dx)...

Math`d/(dt) tan^1(t)=1/(1+t^2)` `y=tan^1(sqrt((1x)/(1+x)))` `y'=(1)/(1+((1x)/(1+x))) * d/(dx)sqrt((1x)/(1+x))` `y'=((1+x)/(1+x+1x)) * d/(dx) (1x)^(1/2) (1+x)^(1/2)` `y'=((1+x)/2) * ((1x)^(1/2)...

MathNote: 1) If y = cosx ; then dy/dx = sinx 2) If y = k ; where k = constant ; then dy/dx = 0 3) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 4) If y...

MathNote: 1) If y = sinx ; then dy/dx = cosx 2) If y = e^x ; then dy/dx = e^x 3) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 4) If y = x^n ; where 'k'...

MathFind `(dy)/(dx) ` if `e^(x/y)=xy ` by implicit differentiation: Note that if u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx) ` . Then: `d/(dx) e^(x/y)=d/(dx)(xy) ` `e^(x/y)...

Math`sqrt(x+y)=1+x^2y^2` `dy/dxsqrt(x+y)=dy/dx(1+x^2y^2)` `1/2sqrt(x+y)^(1/2)(1+dy/dx)=x^2(2y)dy/dx+y^2(2x)` `1/(2sqrt(x+y))+(dy/dx)/(2sqrt(x+y))=2x^2ydy/dx+2xy^2`...

Math`tan^1(x^2y)=x+xy^2` taking derivative on both the sides `d/(dx) tan^1(x^2y)=d/(dx) (x+xy^2)` `1/(1+(x^2y)^2) * d/(dx) (x^2y)=1+x(2y)dy/dx +y^2` `1/(1+x^4y^2) *(x^2 dy/dx+y(2x)) =1+2xy dy/dx...

MathDifferentiate both sides of the equation with respect to x. `d/dx(xsin(y)) + d/dx(ysin(x)) = d/dx 1` Each of these terms needs to be differentiated with the product rule. `x d/dx(siny) + sin(y)d/dx...

MathNote: 1) If y = cosx ; then dy/dx = sinx 2) If y = e^x ; then dy/dx = e^x 3) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 4) If y = sinx ; then...

Math`d/(dx) tan(xy) =d/(dx)(y/(1+x^2))` `sec^2(xy) *d/(dx)(xy) = d/(dx) y(1+x^2)^1` `(1dy/dx)sec^2(xy) = y(1)(1+x^2)^(2)(2x) +(1+x^2)^1 dy/dx` `(1dy/dx)sec^2(xy) = (2xy)/(1+x^2)^2 +...

MathNote: 1) If y = sinx ; then dy/dx = cosx 2) If y = cosx ; then dy/dx = sinx 3) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 4) If y = k ; where...

MathNote: 1) If y = sinx ; then dy/dx = cosx ; 2) If y = u + v ; where both u & v are functions of 'x' , then dy/dx = (du/dx) + (dv/dx) 3) If y = k ; where 'k' = constant ; then dy/dx = 0 4)...

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number 2) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = v*(du/dx) + u*(dv/dx) 3) If y = k ; where 'k' =...

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number 2) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 3) If y = k ; where 'k' =...

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number 2) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 3) If y = k ; where 'k' =...

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number 2) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 3) If y = k ; where 'k' =...

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number 2) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 3) If y = k ; where 'k' =...

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number 2) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 3) If y = k ; where 'k' =...

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number 2) If y = tan^(1)x ; the dy/dx = 1/{1 + (x^2)} Now, the given function contains subfunctions, so the chain rule of...

MathNote: 1) If y = tan^(1)x ; then dy/dx = 1/{1+(x^2)} 2) If y = x^n ; then dy/dx = n*x^(n1) ; where n = real number Clearly, the given function contains subfunctions, so the chain rule of...

Math`x^3+y^3=1` differentiating with respect to x. `3x^2+3y^2(dy/dx)=0` `dy/dx=x^2/y^2`

Math`2sqrt(x)+sqrt(y)=3` differentiating with respect to x. `2(1/(2sqrt(x)))+(1/(2sqrt(y)))(dy/dx)=0` `dy/dx=((2sqrt(y))/sqrt(x))`

Math`x^2+xyy^2=4` Differentiating with respect to x. We get `2x+(y+x(dy/dx))2y(dy/dx)=0` `(2x+y)+(x2y)(dy/dx)=0` `dy/dx=(2yx)/(2x+y)`

MathNote: 1) If y = x^n ; then dy/dx = n*x^(n1) ; where 'n' = real number 2) If y = u*v ; where both u & v are functions of 'x' ; then dy/dx = u*(dv/dx) + v*(du/dx) 3) If y = k ; where 'k' =...

Math`x^4(x+y)=y^2(3xy)` `x^5+x^4y=3xy^2y^3` Differentiating with respect to x.We get `5x^4+(4x^3y+x^4(dy/dx))=(3y^2+3x(2y)(dy/dx))3y^2(dy/dx)` `(5x^4+4x^3y3y^2)=(6xy3y^2x^4)(dy/dx)`...

MathNote: 1) If y = e^x ; then dy/dx = e^x 2) If y = x^n ; then dy/dx = n*x^(n1) ; where 'n' = real number 3) If y = u*v ; where both u & v are functions of 'x' ; then dy/dx = u*(dv/dx) +...

MathNote: 1) If y = cosx ; then dy/dx = sinx 2) If y = x^2 ; then dy/dx = 2x 3) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) Now, the given function is...

MathNote: 1) If y = cosx ; then dy/dx = sinx 2) If y = sinx ; then dy/dx = cosx 3) If y = u*v ; where both u & v are functions of 'x' , then dy/dx = u*(dv/dx) + v*(du/dx) 4) If y = k ; where 'k'...

MathNo, sin^2 (x) is the square of sin(x), or sin(x) taken to the second power. By definition, this means sin(x) multiplied by itself two times: `sin^2(x) = sin(x)*sin(x)` . For example, if x = 30...

MathSlope of a tangent line to a function at a point is equal to the derivative of the function at that point. `y=(1+x^3)^(1/2)` `dy/dx=(1/2)*(1+x^3)^((1/2)1)*3x^2` `dy/dx=(3x^2)/(2(sqrt(1+x^3)))`...

Math`y = pi  x` is the exact answer. At `x = pi` `` `y = sin(sin(pi)) = sin(0) = 0` `y' = 1/2 (cos(x  sin(x)) + cos(x + sin(x))) = 1/2(cos(pi) + cos (pi))` =1. So y = x + pi, using the slope...

Mathy = x `y' = cos(x) + 2sin(x)cos(x).` x = 0 `y' = 1 + 0 = 1` Substituting into the yintercept form of the line, `y = 1x + 0 = x`

MathLet `u = sin(pi*x)` `y = 2^(u), (dy)/(du) = 2^(u)log2` `(du)/(dx) = cos(pi*x) * pi` and `y' = pi*log(2)*2^(sin(pi*x))*cos(pi*x)`

Math`y=x^2e^(1/x)` Derivative can be found by using the product rule `y'= x^2*(e^(1/x)(1*1*x^2)) +e^(1/x)*2x` `y'= x^2(x^2e^(1/x) )+ 2xe^(1/x)` `y' = e^(1/x) +2xe^(1/x)` `y'=(1+2x)e^(1/x)`

Math`d/(dy) [cos((1e^(2x))/(1+e^(2x)))] = sin((1e^(2x))/(1+e^(2x)))[d/(dx) ((1e^(2x))/(1+e^(2x)))]` ` ` `= sin((1e^(2x))/(1+e^(2x))){((1+e^(2x))(d/dx)(1e^(2x)) ...

Math`y=sqrt(1+xe^(2x))` `y'=(1/2)*((1+xe^(2x))^((1/2)1)) *d/dx (sqrt(1+xe^(2x)))` `y'=(1/(2sqrt(1+xe^(2x)))) *(xd/dx e^(2x) +e^(2x)d/dx x)` `y'=(1/(2sqrt(1+xe^(2x)))) *(xe^(2x)*(2) +...