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MathGiven: `y=2x^3+6x^21` `` The derivative is: `y'=6x^2+12x`

MathGiven: `y=(pi)/(2)sin(theta)cos(theta)` The derivative is: `y'=(pi)/(2)cos(theta)+sin(theta)` ``

MathGiven: `g(t)=picos(t)` Find the derivative using the product rule: `d/dx[f(x)g(x)]=f(x)g'(x)+g(x)f'(x)` The derivative is: `g'(t)=pi(sin(t))+cos(t)(0)` ` g'(t)=pisin(t)` ``

MathGiven: `y=x^2(1)/(2)cos(x)` The derivative is: `y'=2x+(1)/(2)sin(x)` ``

MathGiven: `y=7+sin(x)` The derivative is: `y'=cos(x)` ``

MathRewrite the `1/x` term as a fraction. `y= x^1 3sin(x)` Take the derivative of the first term using the power rule. `y'=x^2 3cos(x)` Simplify and eliminate the negative exponent. The answer...

MathGiven: `y=(5)/(2x)^3+2cos(x)` `` Simplify the given. `y=(5)/(8x^3)+2cos(x)` `y=(5x^3)/(8)+2cos(x)` Find the derivative. `y'=(15x^4)/(8)2sin(x)` `y'=(15)/(8x^4)2sin(x)`

MathRewrite the function so that we can use the power rule to take the derivative. `f(x)= 8x^2` Take the derivative. `f'(x)= 2(8) x^3` `f'(x)= 16/x^3` Substitute the given x value from the...

MathRewrite the given function so that we can proceed with the power rule instead of the quotient rule. `f(t)=24t^(1)` Take the derivative of the given function. The derivative of a constant is...

MathTake the derivative for the derivative function. `f'(x)= 21/5 x^2` Substitute the point where x=0 to find the slope at that point. `f'(x)= 21/5 (0)^2= 0` The slope at that point is zero....

MathTake the derivative of the function. `y' = 8x^3` Substitute the x value of the given point. `y'(1) = 8(1)^3 = 8` The derivative, or slope, at point (1,1) is eight. Check by graphing the...

MathUse the power rule and the chain rule to find the derivative. `y' = 2(4x+1) * (4)` `y'=8(4x+1)` `y'=32x+8` Substitute the value at x=0 from the given point. `y'=32(0)+8= 8` The slope at the...

MathTake the derivative of the given function. The chain rule will be shown in the steps. The derivative of the inner function (x4) with the respect to x is 1. `f'(x) =2 * 2(x4)^(21) * (1)`...

MathTake the derivative. Be careful that the derivative of theta with respect to theta is one! `f'(theta)= 4cos(theta)1` Substitute the point given to determine the derivative: ` ` `4cos(0)1=...

MathGiven: `y=12` The derivative of a constant will always be zero: The derivative is: `y'=0` ``

MathGiven: `f(x)=9` The derivative of a constant will always be zero. The derivative is: `f'(x)=0` ``

MathGiven: `y=x^7` The derivative is: `y'=7x^6` ``

MathGiven: `y=x^(12)` The derivative is: `y'=12x^11` ``

MathThis function is the same as: `y= x^5` Use the power rule. `d/dx x^n = nx^(n1)` `y'=5x^(6)` The answer is: `y=5/x^6`

MathGiven: `y=(3)/(x^7)` The given is equivalent to: `y=3x^(7)` Find the derivative: `y'=21x^(8)` Simplify the derivative: `y'=(21)/(x^8)` ``

MathRewrite the function: `y=x^(1/5)` Use the power rule. `y'=1/5 x^(1/51) = 1/5 x^ (4/5) ` The answer is: `y=1/(5x^(4/5))`

MathThe square root of a function can be rewritten as a fraction. `g(x)= x^(1/4)` Use the power rule. `g'(x)= 1/4 x^ (1/41)= 1/4 x^(3/4)` Rewrite the answer to eliminate the negative exponent. The...

MathGiven: `f(x)=x+11` The derivative is: `f'(x)=1` ``

MathGiven: g(x)=6x+3 g'(x)=6 The derivative is: 6

Mathan almost empty bobbin is pulled along a flat surface by a thread which is wrapped around it. The...So we know that a bobbin has an inside wheel that has a diameter of 5 cm and an outer wheel with a diameter of 10 cm. In order to determine how far the end of the thread has moved, we have to...

Mathplease help (1 image)Denote the function under integral as a(t) and its antiderivative as A(t): `a(t) = 3sqrt(t)` , `A'(t) = a(t)` Then `F(x) = A(sin(x))  A(0)` and therefore `F'(x) = A'(sin(x))*[sin(x)]' =...

MathFind the nth term of the quadratic sequences In the form n^2 + b 1) 8, 11, 16, 23, 32 2) 0, 3, 8,...Find the nth term of the quadratic sequences The formulae for quadratic sequences are established by taking second differences between the list of terms. If the sequence is indeed quadratic (of...

MathTo generate terms in a sequence, you would put in the number 1, 2, 3, 4, 5, . . ., etc., for however many terms you are looking for. For instance, to generate the first terms in each sequence, you...

MathThe definite integral `int_0^1 dx/(1+sqrt x)^4` has to be determined. This can be done using substitution. Start with the indefinite integral `int dx/(1+sqrt x)^4` Let `1 + sqrt x = y` `dy/dx =...

MathThe definite integral `int_a^5 x^2+4x+1 dx = 132` . The value of a needs to be determined. `int_a^5 x^2 + 4x + 1 dx` = `int_a^5 x^2 dx+ int_a^5 4x dx+ int_a^5 1 dx ` = `[x^3/3]_a^5 + [4x^2/2]_a^5...

MathThe integral `int (x^2+1)(x^3+3x)^4 dx` has to be determined. Using substitution is the simplest way to arrive at the result. Let `y = x^3+3x` Taking the derivative of both the sides `dy/dx =...

MathIt is not clear from the question if it is n2 or n^2. I am assuming it is n^2. Bronze series: first 5 terms n2 + 1: take n = 1, we get 2. Similarly for n= 2,3,4,5; we get 5,10,17,26. Similarly, for...

MathIf you are asking how to tell if an answer is viable; a viable answer, means that our result is a possible solution to the question being asked. After solving a question or an equation in...

MathThe diameter is an important parameter of a circumference. Geometrically, the diameter is defined as a secant line to the circumference that passes through its center. Another important parameter...

MathIf you are given a fraction you can find an equivalent fraction by multiplying the numerator and denominator by the same number. Example 1: If you are given the fraction 1/2. You can multiply the...

MathA mixed number consists of an integer part and a fraction. In fact the mixed number can be seen as the sum of the integer part plus the fractional part. For example, the mixed number 3 (1/5), can...

MathAnother way to show that `lim_(x>0) sinx/x = 1` is by using the L'Hospital rule. According to this rule, `lim f(x)/g(x) = lim (f'(x))/(g'(x))` . That is, the limit of the ratios of functions...

MathBecause the variable is raised to a variable power, you need to apply the natural logarithm both sides, such that: `ln y = ln (x^(sin x))` Using the property of logarithms, yields: `ln y = sin x*ln...

MathDifferentiate ` y=sqrt(x)^x ` : Rewrite the right hand side: `y=x^(x/2) ` (Note that `sqrt(x)=x^(1/2) ` so `sqrt(x)^x=(x^(1/2))^x=x^(x/2) ` ) Take the natural logarithm of both sides:...

MathDifferentiate `y=(cos(x))^x ` : Take the natural logarithm of both sides: `lny=ln(cos(x))^x ` Use the power property of logarithms: `lny=xln(cos(x)) ` Differentiate; use the product rule on the...

Mathgiven `y = (sinx)^(lnx)` taking log to the base 'e' both sides we get, `lny = lnx*(lnsinx)` Differentiating both sides we get `(1/y)*dy/dx = (1/x)*{lnsinx} + (cosx/sinx)*lnx` or, `y*dy/dx...

Math`y = (tanx)^(1/x)` taking log to the base 'e' both sides we get lny = (1/x)*tanx differentiating both sides we get (1/y)*dy/dx = `[(1/x)*sec^2x] + [(1/x^2)*tanx]` `or, dy/dx = y*[{(1/x)*sec^2x} ...

Mathgiven: `y = (lnx)^cosx` taking log both sides we get `lny = cosx*{ln(lnx)}` Differentiating both sides we get `(1/y)*dy/dx = sinx*{ln(lnx)} + {cosx/(x*lnx)}` `or, dy/dx = y*[sinx*{ln(lnx)} +...

Math`y = ln{(e^x) + x*(e^x)}` taking antilog both sides we get `e^y = (e^x) + x*(e^x)` differentiating both sides `(e^y)dy/dx = (e^x) x*(e^x) + (e^x)` `dy/dx = (e^y)*[x*(e^x)]` `or, dy/dx =...

MathWe have H(z)=ln(sqrt((a^2+z^2)/(a^2z^2))). We will do the differentiation in steps, using the chain rule. Suppose you have H(z)=f(g(u(z))). Then H'(z)=f'(g(u(z)))*g'(u(z))*u'(z). In our case f is...

MathGiven `y=2*x*log10(sqrt(x))` we wish to find y'(x). Write y into separate functions as follows: let `f(x)=2*x, g(x)=log10(x), w(x)=sqrt(x).` Use the Chain rule: `d/dx F(G(x))= F'(G(x)) * G'(x)` and...

MathWe are given y=log2(exp(x)*cos(pi*x))=1/ln(2)*ln(exp(x)*cos(pi*x)). Since the domain of ln(X) is X>0, it means cos(pi*x) cannot equal to zero, we will use this later. First we use the Chain...

MathFind y' and y''. For the first derivative, use the product rule AB'+BA'. Be warned that since we have a ln(2x) term, we will need to use chain rule, which is the derivative of the inner term...

MathYou need to find the first derivative of the function, using the quotient rule, such that: `y' = ((ln x)'*x^2  ln x*(x^2)')/((x^2)^2)` `y' = (x^2/x  2x*lnx)/(x^4)` `y' = (x  2x*lnx)/(x^4)` You...

MathYou need to find the first derivative of the function, using the chain rule, such that: `y' = (ln(x+sqrt(1+x^2)))'` `y' = (ln'(x+sqrt(1+x^2)))*(x+sqrt(1+x^2))'` `y' =...