High School Math : Finding Derivative of a Function

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #15 : Derivatives

What is the first derivative of \(\displaystyle 4x^2+\frac{2}{3}x\)?

Possible Answers:

\(\displaystyle 8\tfrac{2}{3}x^2\)

\(\displaystyle 8x+\frac{2}{3}\)

\(\displaystyle 8\)

\(\displaystyle 3\)

\(\displaystyle 3x\)

Correct answer:

\(\displaystyle 8x+\frac{2}{3}\)

Explanation:

To find the first derivative for this problem, we can use the power rule. The power rule states that we lower the exponent of each of the variables by one and multiply by that original exponent.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+\frac{2}{3}x)=(2*4x^{2-1})+(1*\frac{2}{3}x^{1-1})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+\frac{2}{3}x)=(2*4x^{1})+(1*\frac{2}{3}x^{0})\)

Remember that anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+\frac{2}{3}x)=(8x)+(1*\frac{2}{3}(1))\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+\frac{2}{3}x)=8x+\frac{2}{3}\)

Example Question #1 : General Derivatives And Rules

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+5x-3)=?\)

Possible Answers:

\(\displaystyle 8x-8\)

\(\displaystyle 2x\)

\(\displaystyle 8x+5\)

\(\displaystyle 16x+5\)

\(\displaystyle 3\)

Correct answer:

\(\displaystyle 8x+5\)

Explanation:

This problem is best solved by using the power rule. For each variable, multiply by the exponent and reduce the exponent by one:

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+5x-3)=(2*4x^{2-1})+(1*5x^{1-1})-(0*3x^0-1)\)

Treat \(\displaystyle 3\) as \(\displaystyle 3x^0\) since anything to the zero power is one.

Remember, anything times zero is zero.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+5x-3)=(8x^{1})+(5x^{0})-0\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(4x^2+5x-3)=8x+5\)

Example Question #4 : General Derivatives And Rules

Give the average rate of change of the function \(\displaystyle f(x) = 4 ^{x}\) on the interval  \(\displaystyle [3,4]\).

Possible Answers:

\(\displaystyle 4\)

\(\displaystyle 192\)

\(\displaystyle 64\)

\(\displaystyle 256\)

\(\displaystyle 16\)

Correct answer:

\(\displaystyle 192\)

Explanation:

The average rate of change of \(\displaystyle f\) on interval \(\displaystyle [a,b]\) is 

\(\displaystyle \frac{f (b ) - f(a)}{b-a}\)

Substitute:

\(\displaystyle \frac{f (4 ) - f(3)}{4-3} = \frac{4^{4} - 4^{3}}{1} = \frac{256 - 64}{1} = 192\)

Example Question #17 : Derivatives

What is the derivative of \(\displaystyle x^2+5x\)?

Possible Answers:

\(\displaystyle \frac{1}{3}x^3+\frac{5}{2}x^2+c\)

\(\displaystyle \frac{1}{2}x+5\)

\(\displaystyle x^2\)

\(\displaystyle 2x+5\)

\(\displaystyle 7\)

Correct answer:

\(\displaystyle 2x+5\)

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=(2*x^{2-1})+(1*5x^{1-0})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=2x+5x^0\)

Remember that anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=2x+5*1\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^2+5x)=2x+5\)

Example Question #5 : General Derivatives And Rules

What is the derivative of \(\displaystyle 5x+8\)?

Possible Answers:

\(\displaystyle 13\)

\(\displaystyle 0\)

\(\displaystyle 5\)

\(\displaystyle 5x^2+8x+c\)

\(\displaystyle \frac{5}{2}x^2+8x\)

Correct answer:

\(\displaystyle 5\)

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

We're going to treat \(\displaystyle 8\) as \(\displaystyle 8x^0\), as anything to the zero power is one.

That means this problem will look like this:

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(5x+8)=(5*x^{1-1})+(0*8x^{0-1})\)

Notice that \(\displaystyle (0*8x^{0-1})=0\), as anything times zero is zero.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(5x+8)=(5*x^{1-1})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(5x+8)=(5*x^{0})\)

Remember, anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(5x+8)=5*1\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(5x+8)=5\)

Example Question #2 : Finding Derivatives

What is the derivative of \(\displaystyle x^3+2x+5\)?

Possible Answers:

\(\displaystyle 5\)

\(\displaystyle \frac{1}{3}x^2+\frac{1}{2}x+\frac{1}{5}\)

\(\displaystyle 25\)

\(\displaystyle 3x^2+2x\)

\(\displaystyle 3x^2+2\)

Correct answer:

\(\displaystyle 3x^2+2\)

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

We're going to treat \(\displaystyle 5\) as \(\displaystyle 5x^0\), as anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{3-1})+(1*2x^{1-1})+(0*5x^{0-1})\)

Notice that \(\displaystyle (0*5x^{0-1})=0\), as anything times zero is zero.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{3-1})+(1*2x^{1-1})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{2})+(2x^{0})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=3x^2+2\)

Example Question #3 : Finding Derivatives

What is the derivative of \(\displaystyle 2x\)?

Possible Answers:

\(\displaystyle x\)

\(\displaystyle 2\)

\(\displaystyle \frac{3}{2}x\)

\(\displaystyle \frac{2}{3}x^3+c\)

\(\displaystyle 2+c\)

Correct answer:

\(\displaystyle 2\)

Explanation:

To get \(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(2x)\), we can use the power rule.

Since the exponent of the \(\displaystyle x\) is \(\displaystyle 1\), as \(\displaystyle 2x=2x^1\), we lower the exponent by one and then multiply the coefficient by that original exponent:

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(2x)=1*2x^{1-1}\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(2x)=2x^{0}\)

Anything to the \(\displaystyle 0\) power is \(\displaystyle 1\).

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(2x)=2*1\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(2x)=2\)

Example Question #21 : Derivatives

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(6x^2+8)=?\)

Possible Answers:

\(\displaystyle 12x\)

\(\displaystyle 8\)

\(\displaystyle 0\)

\(\displaystyle 3x\)

\(\displaystyle 20x\)

Correct answer:

\(\displaystyle 12x\)

Explanation:

To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

We're going to treat \(\displaystyle 8\) as \(\displaystyle 8x^0\) since anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(6x^2+8)=(2*6x^{2-1})+(0*8x^{0-1})\)

Notice that \(\displaystyle 0*8x^{0-1}=0\) since anything times zero is zero.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(6x^2+8)=(2*6x^{2-1})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(6x^2+8)=(2*6x^{1})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(6x^2+8)=12x\)

Example Question #1163 : Ap Calculus Ab

What is the derivative of \(\displaystyle x+1\)?

Possible Answers:

\(\displaystyle 0\)

\(\displaystyle 1\)

\(\displaystyle 2x\)

\(\displaystyle \frac{1}{2}x^2\)

\(\displaystyle \frac{1}{x}\)

Correct answer:

\(\displaystyle 1\)

Explanation:

To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

We're going to treat \(\displaystyle 1\) as \(\displaystyle 1x^0\) since anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x+1)=(1*x^{1-1})+(0*x^{0-1})\)

Notice that \(\displaystyle (0*x^{0-1})=0\) since anything times zero is zero.

That leaves us with \(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x+1)=(1*x^{1-1})\).

Simplify.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x+1)=(1*x^{1-1})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x+1)=(x^{0})\)

As stated earlier, anything to the zero power is one, leaving us with:

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(x+1)=1\)

Example Question #5 : Finding Derivative Of A Function

What is the derivative of \(\displaystyle 12x^2+13x+4\)?

Possible Answers:

\(\displaystyle 24x+1\)

\(\displaystyle 2x+1\)

\(\displaystyle 24\)

\(\displaystyle 6x^2+13\)

\(\displaystyle 24x+13\)

Correct answer:

\(\displaystyle 24x+13\)

Explanation:

To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

We're going to treat \(\displaystyle 4\) as \(\displaystyle 4x^0\) since anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(12x^2+13x+4)=(2*12x^{2-1})+(1*13x^{1-1})+(0*4x^{0-1})\)

Notice that \(\displaystyle (0*4x^{0-1})=0\) since anything times zero is zero.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(12x^2+13x+4)=(2*12x^{2-1})+(1*13x^{1-1})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(12x^2+13x+4)=(2*12x^{1})+(1*13x^{0})\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(12x^2+13x+4)=(24x^{1})+(13x^{0})\)

Just like it was mentioned earlier, anything to the zero power is one.

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(12x^2+13x+4)=(24x)+(13*1)\)

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}(12x^2+13x+4)=24x+13\)

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