Derivatives Practice Problems - Power Rule
Problem: Derivative Power Rule #1
Find the derivative f'(x) of:
\Large{f(x) = 5x^3+2x^2-7x+1}
Step 1:
We can use the derivative power rule to find the derivative of the given function. We have:
{f'(x) = 5(3)x^{3-1}+2(2)x^{2-1}-7(1)x^{1-1}}
Notice that we brought down the exponent, and subtracted the exponent by 1 for each term.
Step 2:
Now, after simplifying, we have:
\large{f'(x) = 15x^2+4x-7}
This is our final answer for the derivative.
Solution Complete!
Problem: Derivative Power Rule #2
Find the derivative h'(x) of:
\Large{h(x) = \frac{1}{h^2}} + \frac{3}{h^3}
Step 1:
First, we will remove the fractions, by rewriting the function as follows:
\large{h'(x) = h^{-2} + 3h^{-3}}
Notice that we can now use the derivative power rule to find the derivative.
Step 2:
Applying the power rule, we have:
Problem: Derivative Power Rule #3
Find the derivative P'(x) of:
\Large{P(x) = 6x^6-2x^4+5x^2-1}
Step 1:
We can use the derivative power rule to find the derivative of the given function. We have:
{P'(x) = 6(6)x^{6-1}-2(4)x^{4-1}+5(2)x^{2-1}}
Notice that we brought down the exponent, and subtracted the exponent by 1 for each term.
Step 2:
Problem: Derivative Power Rule #4
Given: \Large{y = 3x^4-\frac{5}{x^3}+2x^2+3}
Find \Large{\frac{dy}{dx}}
Step 1:
First, we will remove the fraction, by rewriting the function as follows:
\large{y = 3x^4-5x^{-3}+2x^2+3}
Notice that we can now use the derivative power rule to find the derivative.
Step 2:
Applying the power rule, we have:
Problem: Derivative Power Rule #5
Given: \Large{y = 7x^5-2x^4+3x^2+1}
Find \Large{\frac{d^2y}{dx^2}}
Step 1:
We can use the derivative power rule to find the derivative of the given function. We have:
{\frac{dy}{dx} = 7(5)x^{5-1}-2(4)x^{4-1}+3(2)x^{2-1}}
Notice that we brought down the exponent, and subtracted the exponent by 1 for each term.
Step 2:
Now, after simplifying, we have:
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