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Questions and Answers
What is the derivative of the function $f(x) = 5x^3$ according to the Power Rule?
What is the result of applying the Constant Rule to the function $f(x) = 10$?
If $f(x) = 2x^2 + 3x  7$, what is $f'(x)$ using the Sum Rule?
Which rule should be applied when differentiating a function expressed as a constant times another function, $f(x) = k g(x)$?
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What is the correct derivative of the polynomial function $f(x) = 4x^5  3x^2 + 7$?
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Which of the following statements about differentiation of polynomials is true?
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What is the derivative of the function $f(x) = 5x^6 + 2x^4 + x$?
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In the process of differentiating the polynomial $f(x) = x^3 + 5x^2  x + 2$, what happens to the term $2$ when differentiated?
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What is the derivative of the term $x^{3/4}$?
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For the polynomial $P(x) = 2x^{1/2} + 4x^{3/5}$, what is $P'(x)$?
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When differentiating $P(x) = 5x^{7/8}  3x^{2/3}$, what is the simplified form of $P'(x)$?
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For a polynomial defined as $P(x) = 4x^{3/2} + 6x^{5/6}$, what approach gives the correct derivative $P'(x)$?
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What is the correct derivative of the polynomial $P(x) = 7x^{5/4}  2x^{2/5} + x^{3/2}$?
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What is the form of the derivative of the cubic curve represented by the function $f(x) = ax^3 + bx^2 + cx + d$?
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If the cubic function is $f(x) = 2x^3  3x^2 + x + 4$, what is the gradient of the tangent at the point $x=1$?
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For the cubic function $f(x) = 3x^3 + 4x^2  x + 5$, what is the general procedure to find the gradient at $x_0$?
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What is the gradient of the normal line for the quadratic function when the gradient of the tangent at point $x_0$ is $3$?
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Given the quadratic function $f(x) = 4x^2 + 2x  5$, what is the gradient of the normal line at $x = 1$?
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If the quadratic function is defined as $f(x) = 3x^2 + 6x + 1$, what is the equation of the normal line at point $x = 2$?
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For the function $f(x) = x^2  4x + 4$, what is the gradient of the normal line at $x = 2$?
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Study Notes
Basic Differentiation: Rules of Differentiation

Power Rule
 If (f(x)=xn)( f(x) = x^n )(f(x)=xn), then (f′(x)=nxn−1)( f'(x) = nx^{n1} )(f′(x)=nxn−1).

Constant Rule
 If ( f(x) = c ) (where ( c ) is a constant), then ( f'(x) = 0 ).

Constant Multiple Rule
 If (f(x)=k⋅g(x))( f(x) = k \cdot g(x) )(f(x)=k⋅g(x)) (where ( k ) is a constant), then (f′(x)=k⋅g′(x))( f'(x) = k \cdot g'(x) )(f′(x)=k⋅g′(x)).

Sum Rule
 If ( f(x) = g(x) + h(x) ), then ( f'(x) = g'(x) + h'(x) ).

Difference Rule
 If ( f(x) = g(x)  h(x) ), then ( f'(x) = g'(x)  h'(x) ).
Differentiation of Polynomials
 Differentiation determines the rate of change of a function.
 Polynomials are functions with terms consisting of constants and variables raised to nonnegative integer powers.
 The power rule states the derivative of (xn)is(nxn−1)( x^n ) is ( n x^{n1} )(xn)is(nxn−1).
 To differentiate a polynomial, apply the power rule to each term individually.
Example
 The derivative of (3x4+2x3−x+5)is(12x3+6x2−1)( 3x^4 + 2x^3  x + 5 ) is ( 12x^3 + 6x^2  1)(3x4+2x3−x+5)is(12x3+6x2−1).
Higher Order Derivatives
 The second derivative is the derivative of the first derivative.
Applications
 Differentiation is widely used in various fields like physics, economics, and engineering.
Key Points
 Differentiate each term in a polynomial separately.
 The derivative of a constant is zero.
 The degree of the polynomial reduces by 1 with each differentiation.
General Tips
 Simplify the fractional exponent before applying the differentiation rules.
 Combine like terms in the derivative for the simplest form.
 Handle negative exponents carefully.
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Description
Test your understanding of the fundamental rules of differentiation in calculus.