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MathGraph the function:

MathGraph the function:

MathGraph the function:

MathGraph the function:

MathGraph the function:

MathGraph both of the functions. g(x): f(x):

MathGraph both function g(x) f(x)

MathGraph the function

MathGraph the function:

MathGraph the function:

MathGraph the function:

MathGraph the function:

MathGraph the function:

MathGraph the function:

Mathsin and arcsin cancel. Leaving the answer to be 0.3

MathTan and arctan cancel This leaves to the answer to be `45 `

Mathcos and arccos cancel. This leaves the answer to be `0.1 `

Mathsin and arcsin cancel. This leaves the answer to be `0.2 `

MathSet `arcsin(sin(3pi))=x` Take the sin of both sides `sin(3pi)=sinx` `0=sinx ` Therefore, `x=0 `

MathYou need to find the absolute extrema of the function, hence, you need to differentiate the function with respect to x, such that: `f'(x) = (2x^3  6x)'` `f'(x) = 6x^2  6` You need to solve for x...

MathYou need to find out the absolute extrema of the given function, hence, you need to differentiate the function with respect to x, such that: `y' = 3*(2/3)*x^(2/3  1)  2` You need to solve for x...

MathYou need to find out the absolute extrema of the given function, hence, you need to differentiate the function with respect to x, such that: `g'(x) = (root(3)(x))'` `g'(x) = (1/3)x^(1/3  1) g'(x)...

MathYou need to evaluate the absolute extrema of the function, hence, you need to differentiate the function with respect to t, using the quotient rule, such that: `g'(t) = ((t^2)'*(t^2 + 3) ...

MathYou need to evaluate the absolute extrema of the function, hence, you need to differentiate the function with respect to x, using the quotient rule, such that: `f'(x) = ((2x)'(x^2 + 1)  2x*(x^2 +...

MathYou need to find the derivative of the function, using the quaotient rule, such that: `h'(s) = (1'*(s  2)  1*(s 2)')/((s2)^2)` `h'(s) = (0*(s  2)  1*1)/((s2)^2)` `h'(s) =1/((s2)^2)` You...

MathYou need to find the derivative of the function h(t), using the quotient rule, such that: `h'(t) = (t'*(t + 3)  t*(t +3)')/((t + 3)^2)` `h'(t) = ((t + 3)  t*1)/((t + 3)^2)` `h'(t) = (t + 3 ...

MathYou need to evaluate the critical numbers of the function and for this reason, you must differentiate the function with respect to x, such that: `f'(x) = (x^3  3x^2)'` `f'(x) = 3x^2  6x` You need...

MathYou need to evaluate the critical numbers of the function and for this reason, you must differentiate the function with respect to x, such that: `f'(x) = (x^4  8x^2)'` `f'(x) = 4x^3  16x` You...

MathYou need to evaluate the critical numbers of the function and for this reason, you must differentiate the function with respect to t, using the product and chain rules, such that: `g'(t) =...

MathGiven: f(x)=3x,[1,2] First find the critical values of the function. To find the critical values of the function (if one exists), set the derivative of the function equal to zero and solve for...

MathGiven: `f(x)=(3/4)x+2,[0, 4]` First find the critical value(s) of the function (if any exist). To find the critical value(s) of the function, set the derivative equal to zero and solve for the...

MathGiven: `h(x)=5x^2, [3, 1]` Find the critical values of the function by setting the derivative equal to zero and solving for x. When the derivative is equal to zero the slope of the tangent line...

Math`f(x) = 4/(x^2 + 1)` `or, f(x) = 4*(x^2 + 1)^1` `thus, f'(x) = 4*(2x)*(x^2 +1)^2` `or, f'(x) = 8x/(x^2 +1)^2` `or, f'(1) = (8*1)/{(1)^2 + 1}^2` `or, f'(x) = 8/4 = 2` ``

Math`f(x) = (3x+1)/(4x3)` `f'(x) = [3*(4x3)  4*(3x+1)]/(4x3)^2` `or, f'(x) = 13/(4x3)^2` `or, f'(4) = 13/(4*4  3)^2` `or, f'(4) = 13/(13)^2` `or, f'(4) = 1/13` ``

Math`y = (1/2)*cosec(2x)` `y' = (1/2)*2*cosec(2x)*cot(2x)` Putting x = pi/4 `y' = 1*cosec(pi/2)*cot(pi/2) = 1*1*0 = 0` ``

Math`y = cosec(3x) + cot(3x)` `y' = 3*cosec(3x)*cot(3x)  3*cosec^2(3x)` Putting x = pi/6 we get y' = `3*cosex(pi/2)*cot(pi/2)  3*cosec^2(pi/2)` `or, y' = 3*1*0  3*1` `or, y' = 3` ``

Math`y = (8x+5)^3` `y' = 3*8*(8x+5)^2` `y' = 24*(8x+5)^2` `y'' = 24*2*8*(8x+5)` `or, y'' = 384*(8x+5)` `` ` `

Math`y = 1/(5x+1) = (5x+1)^1` ` ` `differentiating` `y' = 1{(5x+1)^2}*5` `or, y' = 5(5x+1)^2` ``Differentiating againg w.r.t 'x' we get `y'' = 10*5*(5x+1)^3` `or, y'' = 50/(5x+1)^3` ``

MathNote: 1) If y = cotx ; then dy/dx = `cosec^2(x)` ` ` 2) If y = cosecx ; then dy/dx = cosecx*cotx Now, `f(x) = y = cotx` `differentiating ` `f'(x) = y' = cosec^2(x)` `differentiating` `f''(x) =...

MathNote: If y = sinx ; then dy/dx = cosx If y = cosx ; then dy/dx = sinx 2sinx*cosx = sin(2x) Now, `y = (sinx)^2` `or, y = sin^2x` `differentiating` `y' = 2sinx*cosx` `or,y' = sin(2x)`...

Math`(x^2) + (y^2) = 64` `Differentiating` `2x + 2y(dy/dx) = 0` `or, x + y(dy/dx) = 0` `or, dy/dx = x/y` ``

Math`(x^2) + 4xy  (y^3) = 6` `differentiating` `2x + 4y + 4x(dy/dx)  3(y^2)*(dy/dx) = 0` `or, 2(x+2y) = (dy/dx)*[3(y^2)  4x]` `or, dy/dx = [2(x+2y)]/[3(y^2)4x]` ``

Math`(x^3)*y  x*(y^3) = 4` `differentiating ` `3(x^2)*y + (x^3)*(dy/dx)  (y^3)  3x(y^2)*(dy/dx) = 0` `or, (dy/dx)*[(x^3)3x(y^2)] = (y^3)  3(x^2)y` `or, dy/dx = [(y^3)  3(x^2)y]/[(x^3)3x(y^2)]` ``

MathNote: If y = x^n ; where 'n' = constant ; then dy/dx = n*x^(n1) Now, `(x*y)^(1/2) = x  4y` `or, {x^(1/2)}*{y^(1/2)} = x  4y` `or, (1/2)*{x^(1/2)}*{y^(1/2)} +...

MathNote: 1) If y = sinx; then dy/dx = cosx 2) If y = cos(x) ; then dy/dx = sinx Now, `x*sin(y) = y*cos(x)` `or, x*cosy*(dy/dx) + sin(y) = y*sin(x) + cos(x)*(dy/dx)` `or, x*cos(y)*(dy/dx) ...

MathNote: If y = cos(x) ; then dy/dx = sin(x) Now, `cos(x+y) = x` `or, sin(x+y)*{1 + (dy/dx)} = 1` `or, 1+(dy/dx) = 1/sin(x+y)` `or, 1+(dy/dx) = cosec(x+y)` `or, dy/dx = 1  cosec(x+y)` `or,...

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*{x^(n1)} Now, `y = 15*x^(5/2)` `y' = 15*(5/2)*x^{(5/2)1}` `or, y' = (75/2)*x^(3/2)` `thus, y'' = (75/2)*(3/2)*x^{(3/2)1}` `or, y'' =...

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*{x^(n1)} Now, `y = 20*x^(1/5)` `y' = 20*(1/5)*x^{(1/5)1}` `or, y' = 4*x^(4/5)` `thus, y'' = 4*(4/5)*x^{(4/5)1}` `or, y'' =...

MathNote: If y = tanx ; then dy/dx = sec^2(x) If y = sec(x) ; then dy/dx = sec(x)*tan(x) Now, `f(theta) = 3tan(theta)` `f'(theta) = 3sec^2(theta)` `f''(theta) = 3*2sec(theta)*sec(theta)*tan(theta)`...

MathNote: If y = cos(ax) ; then dy/dx = a*sin(ax) If y = sin(ax) ; then dy/dx = a*cos(ax) ; where 'a' = constant Now, `h(t) = 10cos(t)  15sin(t)` `h'(t) = 10sin(t)  15cos(t)` `h''(t) = 10cos(t)...