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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),...

MathTo 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...

MathRecall 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 antiderivative of f on [a,b]. This...

MathNote 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...

MathMake 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)) ln1 +...

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...

MathIndefinite integral are written in the form of` int f(x) dx = F(x) +C` where: `f(x)` as the integrand ` F(x) ` as the antiderivative function `C` as the arbitrary constant...

MathFor 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...

MathIndefinite 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...

MathBy 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...

MathIndefinite integral are written in the form of` int f(x) dx = F(x) +C` where: f(x) as the integrand F(x) as the antiderivative function C as the arbitrary constant known...

MathTo use logarithmic differentiation, take natural logarithm of both sides: `ln(y) = (x1)ln(x).` Then differentiate this equation with respect to `x` and obtain `(y')/y = (x1)/x + ln(x) = 1  1/x +...

MathFor 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...

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...

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...

MathEquation 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)(xx_0).` The first step to finding equation of tangent line is to calculate the derivative of...

MathThe 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) =...

MathWe 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...

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

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

MathThe 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(x1)/2)` , we apply the derivative for logarithmic functions:...

Math`y=log_10((x^21)/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...

Math`f(x) = log_2 (x^2/(x1))` 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 [...

MathChain 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...

Math`y=log_5 sqrt(x^21)` Before taking the derivative of the function, express the radical in exponent form. `y = log_5 (x^21)^(1/2)` Then, apply the logarithm rule ` log_b (a^m) = m* log_b(a)` . `y...

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...

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

Math`y= log_3(x^23x)` 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)...

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)[...

MathWe 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))'=`...

MathExpress 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...

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) =...

MathThe 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:...

MathRecall 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...

MathWe 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...

Math`y=6^(3x4)` 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)...

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))`...

MathThe 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...

MathTo 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) =...

MathTo simplify the logarithmic equation: `log_5(sqrt(x4))=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...

MathTo 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...

MathIn 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...

Math`log_2 (x1) =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...

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...

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...

MathProblem:`3(5^(x1))=86` ` ` To simplify, we divide both sides by 3: `(3(5^(x1)))/3=(86)/3` `5^(x1)=(86)/3` ` ` ` ` Take the "log" on both sides to apply the logarithm property:...

MathFor exponential equation:`2^(3z)=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:...

MathFor 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:...

MathProblem:` 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...

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 *...