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  • Math
    `arcsin(1/2)` Let this expression be equal to y. `y =arcsin(1/2)` Rewriting it in terms of sine function, the equation becomes: `sin(y) = 1/2` Base on the Unit Circle Chart (see attached figure),...

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    1 educator answer.

  • Math
    To evaluate the integral: `int_(-4)^(4) 3^(x/4) dx` , we follow the formula based from the First Fundamental Theorem of Calculus: `int_a^bf(x)dx=F(b)- F(a)` wherein f is a continuous and F is...

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    1 educator answer.

  • Math
    Recall the First Fundamental Theorem of Calculus: If f is continuous on closed interval [a,b], we follow: `int_a^bf(x)dx` = F(b) - F(a) where F is the anti-derivative of f on [a,b]. This...

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    1 educator answer.

  • Math
    Note that `cos(x) dx = d(sin(x))` and make the substitution `y = sin(x).` The integral becomes `int 2^y dy,` which is (almost) a table one. It is equal to `2^y/ln(2) + C,` which is in turn equal...

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    1 educator answer.

  • Math
    Make the substitution `u = 3^(2x),` then `du = 2ln(3)*3^(2x) dx,` so `3^(2x) dx = (du)/(2 ln(3)).` After this substitution the integral becomes `int(du)/(2 ln(3))/(1 + u) = 1/(2 ln(3)) ln|1 +...

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  • Math
    `int(x+4)6^((x+4)^2)dx=` We will make substitution `u=(x+4)^2.` Differentiation the substitution gives us `du=2(x+4)dx.` Now we rewrite the integral. `1/2 int 6^((x+4)^2)2(x+4)dx=` The above...

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    1 educator answer.

  • Math
    Indefinite integral are written in the form of` int f(x) dx = F(x) +C` where: `f(x)` as the integrand ` F(x) ` as the anti-derivative function `C` as the arbitrary constant...

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  • Math
    For the given integral:` int (x^4+ 5^x) dx` , we may apply the basic integration property: `int (u+v) dx = int (u) dx + int (v) dx` . We can integrate each term separately. `int (x^4+ 5^x) dx =int...

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  • Math
    Indefinite integral is additive, i.e. `int (f+g) = int f + int g.` Therefore `int (x^2 + 2^(-x)) dx =int x^2 dx + int 2^(-x) dx.` Both integrals are table ones or very close to them, so the answer...

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  • Math
    By definition, if the function F(x) is the antiderivative of f(x) then we follow the indefinite integral as `int f(x) dx = F(x)+C` where: f(x) as the integrand F(x) as the...

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  • Math
    Indefinite integral are written in the form of` int f(x) dx = F(x) +C` where: f(x) as the integrand F(x) as the anti-derivative function C as the arbitrary constant known...

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    1 educator answer.

  • Math
    To use logarithmic differentiation, take natural logarithm of both sides: `ln(y) = (x-1)ln(x).` Then differentiate this equation with respect to `x` and obtain `(y')/y = (x-1)/x + ln(x) = 1 - 1/x +...

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  • Math
    For the given problem:` y = x^(2/x)` , we apply the natural logarithm on both sides: `ln(y) =ln(x^(2/x))` Apply the natural logarithm property: `ln(x^n) = n*ln(x)` . `ln(y) = (2/x) *ln(x)` Apply...

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  • Math
    `y = log_10 (2x)` The line is tangent to the graph of function at (5,1). The equation of the tangent line is _______. To solve this, we have to determine the slope of the tangent. Take note that...

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  • Math
    `y=log_3(x)` The line is tangent to the graph of the function at (27,3). The equation of the tangent line is _____. To solve, the slope of the tangent line should be determined. Take note that the...

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  • Math
    Equation of a tangent line to the graph of function `f` at point `(x_0,y_0)` is given by `y=y_0+f'(x_0)(x-x_0).` The first step to finding equation of tangent line is to calculate the derivative of...

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  • Math
    The point is really on the graph, because `2^(-(-1)) = 2.` The tangent line has the slope of `y'(-1),` and the equation `y - 2 = y'(-1)(x+1).` It is clear that `y'(x) = -2^(-x) ln2` and `y'(-1) =...

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    1 educator answer.

  • Math
    We shall use: Product rule `(u cdot v)'=u' cdot v+u cdot v'` Chain rule `(u(v))'=u'(v) cdot v'` `u` and `v` are both functions of arbitrary variable over which we are differentiating. First we...

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  • Math
    The expression under the logarithm `(t/t)` appears to be constant (1), therefore the entire function is a constant, and its derivative is zero.

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  • Math
    First transform the given function using properties of logarithms. Also remember that `xlt1.` `g(x) = log_5(4) - log_5(x^2) - log_5((1-x)^(1/2)) =` `= log_5(4) - 2 log_5(|x|) - 1/2 log_5(1-x).` Now...

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    1 educator answer.

  • Math
    The derivative of a function h with respect to x is denoted by `h'(x)` . To solve for `h'(x)` for given function `h(x) =log_3(x*sqrt(x-1)/2)` , we apply the derivative for logarithmic functions:...

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    1 educator answer.

  • Math
    `y=log_10((x^2-1)/x)` The derivative formula of a logarithm is `d/dx[log_a (u)] = 1/(ln(a) * u)* (du)/dx` Applying this formula, the derivative of the function will be `(dy)/dx = d/dx[log_10...

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    1 educator answer.

  • Math
    `f(x) = log_2 (x^2/(x-1))` The derivative formula of a logarithm is `d/dx [ log_a (u)] = 1/(ln(a) * u) * (du)/dx` Applying this formula, the derivative of the function will be: `f'(x) = d/dx [...

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    1 educator answer.

  • Math
    Chain rule `(u(v))'=u'(v)cdot v'` `u` and `v` are both functions of arbitrary variable over which we are differentiating. We will have to apply the chain rule two times. Once for the logarithm...

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  • Math
    `y=log_5 sqrt(x^2-1)` Before taking the derivative of the function, express the radical in exponent form. `y = log_5 (x^2-1)^(1/2)` Then, apply the logarithm rule ` log_b (a^m) = m* log_b(a)` . `y...

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    1 educator answer.

  • Math
    `g(t) = log_2 (t^2+7)^3` Before taking the derivative of the function, apply the logarithm rule `log_b (a^m)= m*log_b(a)` . So the function becomes: `g(t) = 3log_2(t^2+7)` Take note that the...

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    1 educator answer.

  • Math
    Derivative of a function h with respect to t is denoted as h'(t). The given function: `h(t) = log_5(4-t)^2` is in a form of a logarithmic function. From the derivative for logarithmic functions,...

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    1 educator answer.

  • Math
    `y= log_3(x^2-3x)` The derivative formula of a logarithm is: `d/(dx) [log_a (u)] = 1/(ln(a) * u) * (du)/(dx)` Applying this formula, the derivative of the function will be: `(dy)/(dx) = d/(dx)...

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    1 educator answer.

  • Math
    `y=log_4(5x + 1)` The derivative formula of a logarithm is: `d/(dx) [log_a (u)] = 1/(ln(a) * u) * (du)/(dx)` Applying that formula, the derivative of the function will be: `(dy)/(dx) = d/(dx)[...

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    1 educator answer.

  • Math
    We shall use: Product rule `(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)` Chain rule `(f(g(x)))'=f'(g(x))g'(x)` First we apply product rule. `g'(alpha)=(5^(-alpha/2))'sin(2alpha)+5^(-alpha/2)(sin(2alpha))'=`...

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    1 educator answer.

  • Math
    Express the function `h(theta)` as `u(theta)*v(theta),` where `u(theta) = 2^(-theta)` and `v(theta) = cos(pi theta).` Then we can apply the product rule, `(uv)' = u'v + uv'.` Also we need the...

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    1 educator answer.

  • Math
    `f(t) = 3^(2t)/t` To take the derivative of this function, use the quotient rule `(u/v)'= (v*u' - u*v')/v^2`. Applying that, f'(t) will be: `f'(t) = (t * (3^(2t))' - 3^(2t)*(t)')/t^2` `f'(t) =...

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    1 educator answer.

  • Math
    The derivative of a function f at a point x is denoted as `y' = f'(x)` There are basic properties and formula we can apply to simplify a function such as the Product Rule provides the formula:...

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    1 educator answer.

  • Math
    Recall that the derivative of a function f at a point x is denoted as `y' = f'(x)` . There basic properties and formula we can apply to simplify a function. For the problem `y = x(6^(-2x)),` we...

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    1 educator answer.

  • Math
    We can apply the Product Rule for derivatives: `d/(dx)(u*v) = u' *v + u * v` '. With the given function:` f(x) =x*9^x` , we may let: `u = x` then `u ' = 1` Using the formula for the derivative...

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    1 educator answer.

  • Math
    `y=6^(3x-4)` The derivative formula of an exponential function is: `d/(dx) (a^u) = ln(a) * a^u * (du)/dx` Applying this formula, the derivative of a function will be: `(dy)/(dx) = d/(dx)...

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    1 educator answer.

  • Math
    `y=5^(-4x)` The derivative formula of an exponential function is: `d/(dx) (a^u) = ln(a) * a^u * (du)/(dx)` Applying this formula, the derivative of the function is: `(dy)/(dx) = d/(dx)(5^(-4x))`...

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    1 educator answer.

  • Math
    The derivative of f(x) with respect to x is denoted a f'(x). The given function f(x) = 3^(4x) is in exponential form which means we can apply the formula: `d/(dx)(a^u) = a^u*ln(a)*( du)/(dx)` where...

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    1 educator answer.

  • Math
    To find the derivative, we will use the following formula: `d/(dx)(a^x) = a^x*ln(a)* dx` By comparison, `a^x = 4^x ` then `a=4` , `x=x` , and` d/(dx)(x)=1` Applying the formula, we get: `f'(x) =...

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    1 educator answer.

  • Math
    To simplify the logarithmic equation: `log_5(sqrt(x-4))=3.2` , recall the logarithm property: `a^((log_(a)(x))) = x` . When a logarithm function is raised by the same base, the log cancels out...

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  • Math
    To solve a logarithmic equation, we may simplify or rewrite it using the properties of logarithm. For the given problem `log_(3)(x^2)=4.5` , we may apply the property: `a^((log_(a)(x))) = x` The...

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  • Math
    In solving a logarithmic equation, we may simplify using logarithm properties. Recall the logarithm property: `a^((log_(a)(x))) = x` . When we raise the log with the same base, the "log" will...

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    1 educator answer.

  • Math
    `log_2 (x-1) =5` To solve, convert the equation to exponential form. Take note that if a logarithmic equation is in the form `y = log_b (x)` its equivalent exponential equation is `x= b^y` So...

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    1 educator answer.

  • Math
    `(1+0.1/365)^(365t) = 2` In solving these kind of problems we need to use the logarithm. Take the logarithm on both sides of the equation. `log_(10)((1+0.1/365)^(365t)) = log_10(2)` With logarithms...

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    1 educator answer.

  • Math
    `(1+0.09/12)^(12t) = 3` In solving these kind of problems we need to use the logarithm. Take the log of both sides of the equation. `log_(10)((1+0.09/12)^(12t)) = log_10(3)` With logarithms we know...

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    1 educator answer.

  • Math
    Problem:`3(5^(x-1))=86` ` ` To simplify, we divide both sides by 3: `(3(5^(x-1)))/3=(86)/3` `5^(x-1)=(86)/3` ` ` ` ` Take the "log" on both sides to apply the logarithm property:...

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    1 educator answer.

  • Math
    For exponential equation:`2^(3-z)=625` , we may apply the logarithm property: `log(x^y) = y * log (x)` . This helps to bring down the exponent value. Taking "log" on both sides:...

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    1 educator answer.

  • Math
    For exponential equation: `5^(6x)= 8320` , we may apply the logarithm property: `log(x^y) = y * log (x).` This helps to bring down the exponent value. Taking "log" on both sides:...

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    1 educator answer.

  • Math
    Problem:` 3^(2x)=75` is an exponential equation. To simplify, we need to apply logarithm property: `log(x^y) = y*log(x)` to bring down the exponent that is in terms of x. Taking "log" on both...

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    1 educator answer.

  • Math
    `log_3(x) + log_3(x - 2) = 1` The logarithms at the left side have the same base. So express the left side with one logarithm only using the rule `log_b (M) + log_b (N) = log_b(M*N` ). `log_3(x *...

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