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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`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` ``

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' =...

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

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`

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)` ``

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 = 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: 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 = `(7x +3)^4` `thus, dy/dx = y' = 4{(7x+3)^3}*7` `or, dy/dx = y' = 28(7x+3)^3` ``

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 = 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 = 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 = 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, `h(x) = 6x^2 + 7x^2` `h'(x) = 12x^3 + 14x` `h''(x) = 12*(3)*x^4 + 14` `or, h''(x) = 36x^4 + 14` ``

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` ``

MathYou need to evaluate the derivative of the given function, using the product tule for the products `3x*sin x ` and `x^2*cos x` , such that: `f'(x) = (3x)'*(sin x) + 3x*(sin x)' + (x^2)'(cos x) +...

MathYou need to evaluate the derivative of the given function, using the product rule for the product x*cos x, such that: `f'(x) = ((x)'(cos x) + (x)(cos x)')  (sin x)'` `f'(x) = 1*cos x + x*(sin x)...

MathYou need to evaluate the derivative of the given function, using the product tule for the product `x^2*tan x` , such that: `f'(x) = (2x)'  ((x^2)'(tan x) + (x^2)(tan x)') ` `f'(x) = 2  2x*(tan...

MathYou need to evaluate the derivative of the given function, using the product tule for the product `3x^2*sec x` , such that: `f'(x) = (3x^2)'*(sec x) + 3x^2*(sec x)'` `f'(x) = 6x*(sec x)+ 3x^2*(sec...

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...

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 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 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 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 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'(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) = (5x^2 + 8)'(x^2  4x  6) +...

MathIn order to find the slope of the given function at a specified point, first find the derivative of the function, then evaluate the derivative of the function using the given xvalue. Given:...

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`...

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` ``

MathNote: 1) If y = x^n ; where n = constant ; then dy/dx = n*x^(n1) Now, `f(x) = 27/x^3` `f'(x) = 81/x^4` ``Now, slope of the graph = f'(x) Hence slope at (3,1) = f'(3) = `81/3^4` `or, f'(3) =...

MathGiven: `f(theta)=3cos(theta)(1/4)sin(theta)` The derivative is: `f'(theta)=3sin(theta)(1/4)cos(theta)` ``

MathGiven: `g(alpha)=4cos(alpha)+6` `g'(alpha)=4sin(alpha)` `<br datamcebogus="1">` `<br datamcebogus="1">`

MathGiven: `f(theta)=4theta5sin(theta)` `g'(theta)=45cos(theta)` ``

MathFind the derivative of `h(x)=(8)/(5x^4)` First rewrite the function as `h(x)=(8x^4)/(5)` then find the derivative of the function using the power rule. `h'(x)=(4)(8/5)x^5` The derivative is:...

MathGiven: `g(t)=2/(3t^2)` First rewrite the function as `g(t)=(2t^2)/3` Find the derivative of the function by using the power rule. `g'(t)=2(2/3)t^2` `g'(t)=(4/3)t^3` The derivative is...

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

MathGiven: `h(x)=6sqrt(x)+3root(3)(x)` Rewrite the function as `h(x)=6x^(1/2)+3x^(1/3)` To find the derivative, use the power rule for derivatives. `h'(x)=(1/2)6x^(1/2)+(1/3)3x^(2/3)`...

MathNote: If y = x^n ; where n = constant ; then dy/dx = n*x^(n1) Now, `g(s) = 3s^5  2s^4` `g'(s) = 3*5*s^4  2*4*s^3` `or, g'(s) = 15s^4  8s^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` ``

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` ``

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

MathYou need to find derivative using limit definition, such that: `f'(x)= lim_(Delta x > 0) (f(x + Delta x)  f(x))/(Delta x)` `f'(x) = lim_(Delta x > 0) (6/(x+Delta x)  6/x)/(Delta x)`...

MathYou need to find derivative using limit definition, such that: `f'(x)= lim_(Delta x > 0) (f(x + Delta x)  f(x))/(Delta x)` `f'(x) = lim_(Delta x > 0) ((x + Delta x)^2  4(x+Delta x) + 5 ...

MathYou need to find derivative using limit definition, such that: `f'(x)= lim_(Delta x > 0) (f(x + Delta x)  f(x))/(Delta x)` `f'(x) = lim_(Delta x > 0) (5(x + Delta x)  4  5x + 4)/(Delta...

Math`x^2  y^2 = 36` `or, 2x  2y(dy/dx) = 0` `or, dy/dx = x/y` ``Now, differentiating again we get `2  2y(d^2y/dx^2)  2(dy/dx)^2 = 0` `or, 2  2y(d^2y/dx^2)  2(x^2/y^2) = 0` `or, d^2y/dx^2 = (y^2 ...

Math`x^2 + y^2 = 4` differentiuating both sides w.r.t 'x' we get `2x + 2y(dy/dx) = 0` `or, dy/dx = x/y` Again differentiating w.r.t 'x' we get `2 + 2y{(d^2y)/(dx^2)} + 2(dy/dx)^2 = 0` `or, 1 +...

Math`tan(x+y) = x` `or, sec^2(x+y)*[1 + (dy/dx)] = 1` `or, dy/dx = [1sec^2(x+y)]/{sec^2(x+y)}` ` ` ``Now, dy/dx at (0,0) we get `dy/dx = {1sec^2(0+0)}/{sec^2(0+0)}` `or, dy/dx = (11)/1 = 0` ``

MathGiven: `y^3x^2=4` `3y^2dy/dx2x=0` `3y^2dy/dx=2x` `dy/dx=(2x)/(3y^2)` `dy/dx=(2(2))/(3(2^2))` `dy/dx=4/12=1/3` ``