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Questions and Answers
When differentiating the term $-2x^3$, what is the resulting derivative?
What is the derivative of the polynomial function $f(x) = 2x^5 + 3x^2 - 4$?
Which property of derivatives allows the statement that the derivative of $f(x) + g(x)$ is equal to the derivative of $f(x)$ plus the derivative of $g(x)$?
When calculating the second derivative of the polynomial function $f(x) = x^4 - 6x^2 + 9$, what is the value of $f''(x)$?
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For the polynomial $f(x) = -3x^6 + x^4 + 2x$, which term will contribute $2$ to the derivative?
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Which of the following is a correct derivative calculation for the polynomial $f(x) = 5x^3 - 4x + 7$?
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What is the limit of the expression $\lim_{h \rightarrow 0}\frac{3\sqrt{x + h} - 3\sqrt{x}}{h}$?
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Which expression represents the derivative of the function $y = 3x^{5} - 2x^{3} + 24$?
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What is the instantaneous rate of change for the function $w = \frac{3}{4z^{3}}$?
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Which of the following is the correct formula for the derivative $f^{\prime}(x)$ if $f\left( x \right) = \frac{x^{3} - 3x}{\sqrt{x}}$?
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What is the derivative of the function $g(x) = 3x^{4} + 4\cos x$?
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What does the derivative measure at a specific point on the curve of the function $f\left( x \right) = 4x^{4} + \frac{1}{x^{3}}$?
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Which equation represents the tangent line to the graph of $f\left( x \right) = 4x^{4} + \frac{1}{x^{3}}$ at the point where $x = 1$?
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Which statement correctly describes the Power Rule in differentiation?
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Study Notes
Derivative Rules
-
Power Rule:
- For any function of the form ( y = x^n ), the derivative is given by ( \frac{dy}{dx} = n \cdot x^{n-1} ).
-
Constant Rule:
- The derivative of a constant is zero: ( \frac{d}{dx}(c) = 0 ).
-
Sum/Difference Rule:
- The derivative of a sum/difference is the sum/difference of the derivatives:
- ( \frac{d}{dx}(f(x) + g(x)) = \frac{df}{dx} + \frac{dg}{dx} )
- ( \frac{d}{dx}(f(x) - g(x)) = \frac{df}{dx} - \frac{dg}{dx} ).
- The derivative of a sum/difference is the sum/difference of the derivatives:
Application to the Function ( y = 3x^5 - 2x^3 + 24 )
-
Identify each term:
- ( 3x^5 )
- ( -2x^3 )
- Constant ( 24 )
-
Differentiate each term using the rules:
-
For ( 3x^5 ):
- Apply Power Rule:
- ( \frac{d}{dx}(3x^5) = 3 \cdot 5 \cdot x^{5-1} = 15x^4 )
-
For ( -2x^3 ):
- Apply Power Rule:
- ( \frac{d}{dx}(-2x^3) = -2 \cdot 3 \cdot x^{3-1} = -6x^2 )
-
For the constant ( 24 ):
- Apply Constant Rule:
- ( \frac{d}{dx}(24) = 0 )
-
-
Combine the derivatives:
- Total derivative:
- ( \frac{dy}{dx} = 15x^4 - 6x^2 + 0 )
- Simplified result:
- ( \frac{dy}{dx} = 15x^4 - 6x^2 )
Summary of Derivative
- The derivative of ( y = 3x^5 - 2x^3 + 24 ) is:
- ( \frac{dy}{dx} = 15x^4 - 6x^2 )
Derivative Rules
- The Power Rule states that the derivative of a function of the form ( y = x^n ) is given by ( \frac{dy}{dx} = n \cdot x^{n-1} ).
- The Constant Rule states that the derivative of a constant is zero: ( \frac{d}{dx}(c) = 0 ).
- The Sum/Difference Rule states that the derivative of a sum/difference is the sum/difference of the derivatives:
- ( \frac{d}{dx}(f(x) + g(x)) = \frac{df}{dx} + \frac{dg}{dx} )
- ( \frac{d}{dx}(f(x) - g(x)) = \frac{df}{dx} - \frac{dg}{dx} ).
Applying Derivative Rules to ( y = 3x^5 - 2x^3 + 24 )
- The function ( y = 3x^5 - 2x^3 + 24 ) consists of three terms: ( 3x^5 ), ( -2x^3 ), and the constant ( 24 ).
- To find the derivative of the function, each term is differentiated individually using the appropriate rule.
- Applying the Power Rule to ( 3x^5 ) results in ( \frac{d}{dx}(3x^5) = 15x^4 ).
- Applying the Power Rule to ( -2x^3 ) results in ( \frac{d}{dx}(-2x^3) = -6x^2 ).
- Applying the Constant Rule to ( 24 ) results in ( \frac{d}{dx}(24) = 0 ).
- Combining the derivatives of each term gives ( \frac{dy}{dx} = 15x^4 - 6x^2 + 0 ), which simplifies to ( \frac{dy}{dx} = 15x^4 - 6x^2 ).
Summary of the Derivative
- The derivative of the function ( y = 3x^5 - 2x^3 + 24 ) is ( \frac{dy}{dx} = 15x^4 - 6x^2 ).
Derivatives of Polynomials
- The derivative of a polynomial function measures how the function's value changes as its input changes.
- The derivative of a term (ax^n) is (n \cdot ax^{n-1}), where (a) is a constant and (n) is a non-negative integer.
- To find the derivative of a polynomial, apply the power rule to each term and add the results.
- The derivative of a constant is zero.
- The derivative of a sum is the sum of derivatives, and the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
- The second derivative is the derivative of the first derivative, and it helps determine concavity and points of inflection.
- Derivatives are used for understanding the behavior of polynomial functions, like identifying increasing/decreasing intervals and local extrema, and for solving optimization problems involving finding maximum and minimum values.
- Graphically, the derivative at a point represents the slope of the tangent line to the curve at that point.
Derivatives
- The derivative of a function f(x) is the instantaneous rate of change of f(x) with respect to x.
- The derivative of f at x is denoted by f'(x).
- To find the derivative of a function, take the limit of the difference quotient as h approaches 0, written as:
- [limh→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}limh→0hf(x+h)−f(x)].
Power Rule
- The power rule states that the derivative of x raised to the power of n is equal to n times x raised to the power of n-1:
- [ddxxn=nxn−1\frac{d}{dx} x^n = nx^{n-1}dxdxn=nxn−1].
- This rule applies for any real number n.
Finding Derivatives
- To find the derivative of a function with multiple terms, find the derivative of each term separately.
- For example, the derivative of f(x) = 3x^2 - 4x + 5 is found by taking the derivative of each term:
- [ddx(3x2)=6x\frac{d}{dx}(3x^2) = 6xdxd(3x2)=6x].
- [ddx(−4x)=−4\frac{d}{dx}(-4x) = -4dxd(−4x)=−4].
- [ddx(5)=0\frac{d}{dx}(5) = 0dxd(5)=0].
- The final derivative is f'(x) = 6x - 4.
Tangent Lines
- The derivative of a function f(x) at a point x gives the slope of the tangent line to the graph of f(x) at that point.
- To find the equation of the tangent line, use the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the line.
- The derivative is the slope.
- The point on the line is (x1, f(x1)), where x1 is the x-value at which you are finding the tangent line.
Limits
- A limit is the value that a function approaches as the input approaches some value.
- To evaluate a limit, try substituting the value that the input is approaching into the function.
- If substituting the value causes the function to be undefined, try simplifying the function by factoring or using other algebraic techniques.
- There are many ways to find the derivative from limits.
- Limits are used to find the instantaneous rate of change of a function at a point, to find the slope of the tangent line, and to find the derivative.
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Description
This quiz covers the fundamental derivative rules, including the Power Rule, Constant Rule, and Sum/Difference Rule. It applies these concepts to differentiate a polynomial function step-by-step. Test your understanding of these essential calculus principles.
