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Calculus 1 : How to find solutions to differential equations

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Example Questions

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Example Question #21 :Solutions To Differential Equations

Differentiate the expression.

$e^{6x}$

Possible Answers:

$e^{6x}$

6xe

$6e^{6x}$

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 #22 :Solutions To Differential Equations

Differentiate the expression.

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

Possible Answers:

$2x\sin(x)$

x^2

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

$x\cos(x)$

$2x\cos(x)$

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 #23 :Solutions To 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 #24 :Solutions To Differential Equations

Find {f}'(x) ．

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

Possible Answers:

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

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

$e^{4x}$

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

$4e^{x}$

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 #25 :Solutions To Differential Equations

Find {f}'(x) ．

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

Possible Answers:

2x-4

(x+2)(x-2)

$x^{2}$

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 #26 :Solutions To Differential Equations

Differentiate the logarithm.

$\ln(4x)$

Possible Answers:

$\frac{1}{4x}$

$4\ln(4x)$

$\frac{1}{x}$

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 #21 :Solutions To Differential Equations

Differentiate the polynomial.

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

Possible Answers:

x^2+4x+7

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

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

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

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

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 #28 :Solutions To Differential Equations

Find {f}'(x) ．

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

Possible Answers:

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

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

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

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

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

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 #29 :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 #30 :Solutions To Differential Equations

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

Possible Answers:

y=Ce^2^x

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

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

y=Ce^0^.^5^x

y=Ce^x

Correct answer:

y=Ce^0^.^5^x

Explanation:

Rewrite $\frac{dy}{dx}= \frac{y}{2}$ by multiply theon both sides, and dividingon 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 : How to find solutions to differential equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #21 :Solutions To Differential Equations

Example Question #22 :Solutions To Differential Equations

Example Question #23 :Solutions To Differential Equations

Example Question #24 :Solutions To Differential Equations

Example Question #25 :Solutions To Differential Equations

Example Question #26 :Solutions To Differential Equations

Example Question #21 :Solutions To Differential Equations

Example Question #28 :Solutions To Differential Equations

Example Question #29 :Solutions To Differential Equations

Example Question #30 :Solutions To Differential Equations

All Calculus 1 Resources

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