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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) =...

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)...

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*{x^(n1)} Now, y = `(7x +3)^4` `thus, dy/dx = y' = 4{(7x+3)^3}*7` `or, dy/dx = y' = 28(7x+3)^3` ``

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*{x^(n1)} Now, `y = (x^2  6)^3` `thus, dy/dx = y' = 3*{(x^2  6)^2}*(2x)` `or, dy/dx = y' = 6x*(x^2  6)^2` ``

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

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

MathNote: If y = cos(ax) ; then dy/dx = a*sin(ax) Now, `y = 5cos(9x + 1)` `dy/dx = y' = 5*(sin(9x+1))*9` `or, dy/dx = y' = 45sin(9x+1)` ``

Mathy = 1 cos(2x) + 2`(cos^2x)` `dy/dx = y' = 2*sin(2x)  4*cosx*sinx` `or, y' = 2sin(2x)  2sin(2x) = 0` note: `2sinx*cosx = sin2x`

MathYou need to differentiate the function with respect to x, using the product rule and chain rule, such that: `y' = x'*(6x + 1)^5 + x*((6x + 1)^5)'` `y' = 1*(6x + 1)^5 + x*5*(6x + 1)^4*(6x+1)'` `y' =...

MathIn order to find the slope of a function at a given point, find the derivative of the function then plug in the xvalue. Given: `f(x)=3x^24x, (1, 1)` `f'(x)=6x` `f'(1)=6(1)` `f'(1)=6` ``

MathIn order to find the slope of a function at a specific point, first find the derivative of the function, then plug in the xvalue from the given point. Given: `f(x)=2x^48, (0,8)` `f'(x)=8x`...

MathYou need to evaluate the derivative of the given function and since the function is a product of two polynomials, then you must use the product rule, such that: `f'(x) = (5x^2 + 8)'(x^2  4x  6) +...

MathYou need to evaluate the derivative of the given function and since the function is a product of two polynomials, then you must use the product rule, such that: `f'(x) = (2x^3 + 5x)'(3x4) + (2x^3...

MathYou need to evaluate the derivative of the given function and since the function is a product of two polynomials, then you must use the product rule, such that: `f'(x) = (sqrt x)'(sin x) + (sqrt...

MathYou need to evaluate the derivative of the given function and since the function is a product of two polynomials, then you must use the product rule, such that: `f'(t) = (2t^5)'(cos t) + (2t^5)(cos...

MathYou need to evaluate the derivative of the given function and since the function is a quotient of two polynomials, then you must use the quotient rule, such that: `f'(x) = ((x^2+x1)'(x^21) ...

MathYou need to evaluate the derivative of the given function and since the function is a quotient of two polynomials, then you must use the quotient rule, such that: `f'(x) = ((2x+7)'(x^2+4) ...

MathYou need to evaluate the derivative of the given function and since the function is a quotient of two functions, then you must use the quotient rule, such that: `f'(x) = ((x^4)'(cos x)  (x^4)(cos...

MathYou need to evaluate the derivative of the given function and since the function is a quotient of two functions, then you must use the quotient rule, such that: `f'(x) = ((x^4)(sin x)'  (x^4)'(sin...

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*x^(n1) Now, `g(t) = 8t^3 5t + 12` `g'(t) = 8*3*t^2  5 + 0` `or, g'(t) = 24t^2  5` `Thus, g''(t) = 24*2*t^1  0` `or, g''(t) = 48t` ``

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*x^(n1) Now, `h(x) = 6x^2 + 7x^2` `h'(x) = 12x^3 + 14x` `h''(x) = 12*(3)*x^4 + 14` `or, h''(x) = 36x^4 + 14` ``

MathAs per the rules of differentiation, derivative of a constant is zero Thus, If y = 25 dy/dx = y' = 0

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*x^(n1) Now, ` y = 4t^4` `or, y' = 4*4*t^3` `or, y' = f'(t) = 16*t^3` ``

MathNote: If y = x^n ; then dy/dx = n*x^(n1) ; where n = constant Now, `f(x) = x^3  11x^2` `f'(x) = 3*x^2  11*2*x^1` `or, f'(x) = 3x^2  22x` ``