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Calculus 1 : Differential Equations

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Example Questions

Example Question #31 : Differential Equations

Differentiate the expression.

$e^{6x}$

Possible Answers:

6xe

$6e^{6x}$

$e^{6x}$

Correct answer:

$6e^{6x}$

Explanation:

We will use the fact that $e^{au} = ae^{au}$ to differentiate. Let a=6 and u=x . Substituing our values we can see the derivative will be $6e^{6x}$ .

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Example Question #32 : Differential Equations

Differentiate the expression.

$x^{2}\sin(x)$

Possible Answers:

$x (2 \sin(x)+x \cos(x))$

x^2

$2x\cos(x)$

$2x\sin(x)$

$x\cos(x)$

Correct answer:

$x (2 \sin(x)+x \cos(x))$

Explanation:

Using the product rule, we determine the derivative of $f(x)\cdot g(x) = {f}'(x)\cdot g(x) + f(x)\cdot {g}'(x))$
Let $f(x) = x^{2}$ and $g(x) = \sin(x)$ . We can see that f{}'(x) = 2x and $g{}'(x) = \cos(x)$ .

Plugging in our values into the product rule formula, we are left with the final derivative of $x (2 \sin(x)+x \cos(x))$ .

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Example Question #31 : Differential Equations

Differentiate the value.

Possible Answers:

$74x^{2}$

74x

Correct answer:

Explanation:

According to the power rule, whenever we differentiate a constant value it will reduce to zero. Since the only term of our function is a constant, we can only differentiate $74 \rightarrow 0$ .

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Example Question #34 : Differential Equations

Find {f}'(x) .

$f(x) = e^{2x^{2}}$

Possible Answers:

$4xe^{2x^{2}}$

$e^{2x^{2}}$

$4e^{x}$

$e^{4x}$

$2e^{x^{2}}$

Correct answer:

$4xe^{2x^{2}}$

Explanation:

Using the chain rule, we will differentiate the exponent of our exponential function, and then multiply our original function. Differentiating our exponent with the power rule will yield . Using the chain rule we will multiply this by our original function resulting in $4xe^{2x^{2}}$ .

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Example Question #35 : Differential Equations

Find {f}'(x) .

$f(x) = x^{2}-4$

Possible Answers:

$x^{2}$

(x+2)(x-2)

2x-4

Correct answer:

Explanation:

Using the power rule, we can differentiate our first term reducing the power by one and multiplying our term by the original power. $x^{2}$ , will thus become . The second term is a constant value, so according to the power rule this term will become .

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Example Question #36 : Differential Equations

Differentiate the logarithm.

$\ln(4x)$

Possible Answers:

$\frac{1}{x}$

$4\ln(4x)$

$\frac{1}{4x}$

Correct answer:

$\frac{1}{x}$

Explanation:

Using the chain rule, we will determine the derivative of our function will be $\frac{d\log(u)}{du}\cdot \frac{du}{dx}$ .

The derivative of the log function is $\frac{1}{x}$ , and our second term of the chain rule will cancel out $\frac{1}{4x}\cdot 4=\frac{4}{4x}=\frac{1}{x}$ .

Thus our derivative will be $\frac{1}{x}$ .

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Example Question #32 : Differential Equations

Differentiate the polynomial.

$x^{4}+3x^{3}+4x$

Possible Answers:

$3x^{3}+4x^{2}$

$4x^{3}+9x^{2}+4$

$4x^{3}+9x^{2}$

$x^{3}+3x^{2}+4$

x^2+4x+7

Correct answer:

$4x^{3}+9x^{2}+4$

Explanation:

Using the power rule, we can differentiate our first term reducing the power by one and multiplying our term by the original power. $x^{4}$ , will thus become $4x^{3}$ . The second term $3x^{3}$ , will thus become $9x^{2}$ . The last term is , will reduce to .

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Example Question #37 : Differential Equations

Find {f}'(x) .

$f(x) = \frac{e^x}{x^2}$

Possible Answers:

$\frac{x^2-e^x}{x^2}$

$\frac{e^x (x-2)}{x^3}$

$\frac{e^{x^{2}} (x-2)}{x^3}$

$\frac{(x-2)}{x^2}$

$\frac{(x-2)}{x^3}$

Correct answer:

$\frac{e^x (x-2)}{x^3}$

Explanation:

According to the quotient rule, the derivative of ,

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

We will let $f(x) = e^{x} , f{}'(x) = e^{x} , g(x) = x^{2}$ and g{}'(x) = 2x
Plugging all of our values into the quotient rule formula we come to a final solution of :

$\frac{e^x (x-2)}{x^3}$ .

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Example Question #25 : Solutions To Differential Equations

Differentiate the polynomial.

$x^{3}+3x^{2}$

Possible Answers:

$3x^{2}+6x$

$3x^{3}+6x$

$3x^{4}+6x^{3}$

$x^{3}+3$

Correct answer:

$3x^{2}+6x$

Explanation:

Using the power rule, we can differentiate our first term reducing the power by one and multiplying our term by the original power. $x^{3}$ , will thus become $3x^{2}$ . The second term $3x^{2}$ , will thus become .

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Example Question #33 : Differential Equations

Solve the differential equation: $\frac{dy}{dx}= \frac{y}{2}$

Possible Answers:

y=Ce^0^.^5^x

y=Ce^ -^0^.^5^x

y=Ce^2^x

y=Ce^ -^0^.^2^5^x

y=Ce^x

Correct answer:

y=Ce^0^.^5^x

Explanation:

Rewrite $\frac{dy}{dx}= \frac{y}{2}$ by multiply the on both sides, and dividing on both sides of the equation.

$\frac{1}{y}dy= \frac{1}{2}dx$

Integrate both sides of the equation and solve for y.

$\int\frac{1}{y}dy= \int\frac{1}{2}dx$

$ln\left | y\right |=\frac{1}{2}x+C$

$\left | y\right |= e^0^.^5^xe^C$

y=Ce^0^.^5^x

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Calculus 1 : Differential Equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #31 : Differential Equations

Example Question #32 : Differential Equations

Example Question #31 : Differential Equations

Example Question #34 : Differential Equations

Example Question #35 : Differential Equations

Example Question #36 : Differential Equations

Example Question #32 : Differential Equations

Example Question #37 : Differential Equations

Example Question #25 : Solutions To Differential Equations

Example Question #33 : Differential Equations

All Calculus 1 Resources

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