How to find differential functions - AP Calculus AB

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Question

Find the derivative.

Answer

$\frac{d}{dx}9=0$

Use the power rule to find the derivative.

$\frac{d}{dx}-2x=-2$

Thus, the derivative is

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Question

Find the derivative.

Answer

Use the power rule to find this derivative.

$\frac{d}{dx}18=0$

$\frac{d}{dx}-4x=-4$

$\frac{d}{dx}2x^2=4x$

Thus, the derivative is

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Question

Find the derivative.

Answer

Recall that the derivative of a constant is zero.

$\frac{d}{dx}100=0$

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Question

Find the derivative.

Answer

Use the power rule to find the derivative.

$\frac{d}{dx}x^4=4x^3$

$\frac{d}{dx}-x^3=-3x^2$

$\frac{d}{dx}x=1$

Thus, the derivative is .

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Question

What is the derivative of $f(x)=\frac{4}{3}x^3-5x^2$ ?

Answer

When taking the derivative, remember to multiply the exponent by the coefficient in front of the x term and then subtract one from the exponent. Therefore, after differentiating each term, you get: $f{}'(x)=4x^2-10x$ .

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Question

Find the derivative.

Answer

Use the power rule to find the derivative.

$\frac{d}{dx}5x^2=10x$

$\frac{d}{dx}-6x=-6$

$\frac{d}{dx}3=0$

Thus, the derivative is

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Question

Find the derivative of the following function:

$f(x)= 4e^{2x} + 3x^4$

Answer

If , then the derivative is .

If , then the derivative is $f'(x)= Cx^{C-1}$ .

If , then the derivative is $f'(x)= \frac{1}{x}$ .

If , the the derivative is .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If , then the derivative is . This is known as the chain rule.

In this case, we must find the derivative of the following: $f(x)= 4e^{2x} + 3x^4$

That is done by doing the following: $f'(x)= 4e^{2x}(2) + (4)3x^{4-1}$

Therefore, the answer is: $f'(x)= 8e^{2x} + 12x^{3}$

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Question

Find the derivative:

Answer

If , then the derivative is .

If , then the derivative is $f'(x)= Cx^{C-1}$ .

If , then the derivative is $f'(x)= \frac{1}{x}$ .

If , the the derivative is .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If , then the derivative is . This is known as the chain rule.

In this case, we must find the derivative of the following:

That is done by doing the following: $f'(x)= \frac{1}{2x+e^x} (2x^{1-1}+e^x)$

Therefore, the answer is: $f'(x)= \frac{2+e^x}{2x+e^x}$

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Question

Find the derivative at x=3.

Answer

First, find the derivative using the power rule:

$\frac{d}{dx}6x^3=18x^2$

Now, substitute 3 for x.

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Question

Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.

Answer

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

$f'(c)=\frac{f(b)-f(a)}{b-a}=\frac{Rise}{Run};a<c<b$

Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.

First, find the two function values of on the interval

Then take the difference of the two and divide by the interval.

$\frac{-1944-0}{3-0}=-648$

Now find the derivative of the function; this will be solved for the value(s) found above.

Using a calculator, we find the solution , which fits within the interval , satisfying the mean value theorem.

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Question

Find the derivative.

$\frac{2}{3x^4}$

Answer

Use the quotient rule to find the derivative.

$\frac{3x^4(0)-2(12x^3)}{(3x^4)^2}=\frac{-24x^3}{9x^8}$

Simplify.

The derivative is $\frac{-8}{3x^5}$ .

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Question

Find the derivative.

$\frac{1}{x^5}$

Answer

Use the quotient rule to find the derivative.

$\frac{x^5(0)-1(5x^4)}{(x^5)^2}=\frac{-5x^4}{x^{10}}=\frac{-5}{x^6}$

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Question

Find the derivative.

$12x^{12}$

Answer

Use the power rule to find the derivative.

$\frac{d}{dx}12x^{12}=144x^{11}$

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Question

Differentiate:

$f(x) = \left(\frac{1}{x}\right)\cos^2(x)$

Answer

To find the derivative of this function we must use the Product Rule and the Chain Rule. First we set

$h(x) = \frac{1}{x}$

and

$g(x) = \cos^2(x)$

Now differentiating both of these functions gives

$h'(x) = \frac{-1}{x^2}$

$g'(x) = 2\cos(x)(-\sin(x)) = -2\cos(x)\sin(x)$

Applying this to the Product Rule gives us,

$f'(x) = \frac{-1}{x^2}\cos^2(x)-\frac{2}{x}\cos(x)\sin(x)$

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Question

Let on the interval $(0,\pi)$ . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.

Answer

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

$f'(c)=\frac{f(b)-f(a)}{b-a}=\frac{Rise}{Run};a<c<b$

Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.

First, find the two function values of on the interval $(0,\pi)$

$f(\pi)=4cos(\pi)=-4$

Then take the difference of the two and divide by the interval.

$\frac{-4-4}{\pi-0}=\frac{-8}{\pi}$

Now find the derivative of the function; this will be solved for the value(s) found above.

Trigonometric derivative:

$-4sin(x)=\frac{-8}{\pi}$

Using a calculator, we find the solutions , which fit within the interval $(0,\pi)$ , satisfying the mean value theorem.

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Question

Differentiate the function

$f(x) = \sin(\cos(2x))$

Answer

To differentiate the function properly, we must use the Chain Rule which is,

$f(x) = h(g(x)) \rightarrow f'(x) = h'(g(x))g'(x)$

Therefore the derivative of the function is,

$f'(x) = \cos(\cos(2x))(-\sin(2x))(2) = -2\cos(\cos(2x))\sin(2x)$

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Question

Find the derivative.

$-\frac{5}{x^3}$

Answer

Use the quotient rule to find this derivative.

Recall the quotient rule:

$\ \ h(x)=\frac{f(x)}{g(x)} \ \ h'(x)=\frac{g(x)f'(x)-g'(x)f(x)}{(g(x))^2}$

$h'(x)=\frac{x^3(0)-(-5)3x^2}{x^6}=\frac{15x^2}{x^6}=\frac{15}{x^4}$

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Question

Differentiate

$f(x) = \sin^2(x^2)$

Answer

To differentiate this equation we use the Chain Rule.

$f(x) = h(g(x)) \rightarrow f'(x) = h'(g(x))\cdot g'(x)$

Using this throughout the equation gives us,

$f'(x) = 2\sin(x^2)\cos(x^2)(2x)$

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Question

Find the derivative of

$f(x) = 4\sin(x+3x^2)$

Answer

To find the derivative of the function we must use the Chain Rule, which is

$f(x) = h(g(x)) \rightarrow f'(x) = h'(g(x))\cdot g'(x)$

Applying this to the function we get,

$f'(x) = 4\cos(x+3x^2)(1+6x)$

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Question

Find the slope of the function at .

Answer

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

at

x:

$\frac{\delta f}{\delta x}=12x^2+6xy+2y^2;12(1)^2+6(1)(1)+2(1)^2=20$

y:

$\frac{\delta f}{\delta y}=3x^2+4xy+3y^3;3(1)^2+4(1)(1)+3(1)^3=10$

The slope is

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