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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 derivative of the function $f(x) = 5x^3$ according to the Power Rule?
- $15x^2$ (correct)
- $20x^4$
- $5x^2$
- $3x^4$
What is the result of applying the Constant Rule to the function $f(x) = 10$?
What is the result of applying the Constant Rule to the function $f(x) = 10$?
- $0$ (correct)
- $1$
- $10$
- Undefined
If $f(x) = 2x^2 + 3x - 7$, what is $f'(x)$ using the Sum Rule?
If $f(x) = 2x^2 + 3x - 7$, what is $f'(x)$ using the Sum Rule?
- $4x + 3$ (correct)
- $2x - 3$
- $2x + 3$
- $4x - 3$
Which rule should be applied when differentiating a function expressed as a constant times another function, $f(x) = k g(x)$?
Which rule should be applied when differentiating a function expressed as a constant times another function, $f(x) = k g(x)$?
What is the correct derivative of the polynomial function $f(x) = 4x^5 - 3x^2 + 7$?
What is the correct derivative of the polynomial function $f(x) = 4x^5 - 3x^2 + 7$?
Which of the following statements about differentiation of polynomials is true?
Which of the following statements about differentiation of polynomials is true?
What is the derivative of the function $f(x) = -5x^6 + 2x^4 + x$?
What is the derivative of the function $f(x) = -5x^6 + 2x^4 + x$?
In the process of differentiating the polynomial $f(x) = x^3 + 5x^2 - x + 2$, what happens to the term $2$ when differentiated?
In the process of differentiating the polynomial $f(x) = x^3 + 5x^2 - x + 2$, what happens to the term $2$ when differentiated?
What is the derivative of the term $x^{3/4}$?
What is the derivative of the term $x^{3/4}$?
For the polynomial $P(x) = 2x^{1/2} + 4x^{3/5}$, what is $P'(x)$?
For the polynomial $P(x) = 2x^{1/2} + 4x^{3/5}$, what is $P'(x)$?
When differentiating $P(x) = 5x^{7/8} - 3x^{2/3}$, what is the simplified form of $P'(x)$?
When differentiating $P(x) = 5x^{7/8} - 3x^{2/3}$, what is the simplified form of $P'(x)$?
For a polynomial defined as $P(x) = 4x^{3/2} + 6x^{5/6}$, what approach gives the correct derivative $P'(x)$?
For a polynomial defined as $P(x) = 4x^{3/2} + 6x^{5/6}$, what approach gives the correct derivative $P'(x)$?
What is the correct derivative of the polynomial $P(x) = 7x^{5/4} - 2x^{2/5} + x^{3/2}$?
What is the correct derivative of the polynomial $P(x) = 7x^{5/4} - 2x^{2/5} + x^{3/2}$?
What is the form of the derivative of the cubic curve represented by the function $f(x) = ax^3 + bx^2 + cx + d$?
What is the form of the derivative of the cubic curve represented by the function $f(x) = ax^3 + bx^2 + cx + d$?
If the cubic function is $f(x) = 2x^3 - 3x^2 + x + 4$, what is the gradient of the tangent at the point $x=1$?
If the cubic function is $f(x) = 2x^3 - 3x^2 + x + 4$, what is the gradient of the tangent at the point $x=1$?
For the cubic function $f(x) = 3x^3 + 4x^2 - x + 5$, what is the general procedure to find the gradient at $x_0$?
For the cubic function $f(x) = 3x^3 + 4x^2 - x + 5$, what is the general procedure to find the gradient at $x_0$?
What is the gradient of the normal line for the quadratic function when the gradient of the tangent at point $x_0$ is $3$?
What is the gradient of the normal line for the quadratic function when the gradient of the tangent at point $x_0$ is $3$?
Given the quadratic function $f(x) = 4x^2 + 2x - 5$, what is the gradient of the normal line at $x = 1$?
Given the quadratic function $f(x) = 4x^2 + 2x - 5$, what is the gradient of the normal line at $x = 1$?
If the quadratic function is defined as $f(x) = -3x^2 + 6x + 1$, what is the equation of the normal line at point $x = 2$?
If the quadratic function is defined as $f(x) = -3x^2 + 6x + 1$, what is the equation of the normal line at point $x = 2$?
For the function $f(x) = x^2 - 4x + 4$, what is the gradient of the normal line at $x = 2$?
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^{n-1} )(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 non-negative integer powers.
- The power rule states the derivative of (xn)is(nxn−1)( x^n ) is ( n x^{n-1} )(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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Test your understanding of the fundamental rules of differentiation in calculus.
