32 Questions
What is the derivative of the constant function f(x) = 7?
0
According to Theorem 3.3.2, what is the derivative of $x^4$?
$4x^3$
If f(x) = 3, what is the derivative f'(x)?
0
What is the derivative of the function g(x) = 2x?
2x
According to Theorem 3.3.1, what is the derivative of the constant function f(x) = -9?
0
What is the derivative of the function h(x) = 5x^2?
$10x$
What is the derivative of f(x) + g(x) with respect to x?
f'(x) + g'(x)
According to Theorem 3.3.5, what is the derivative of f(x) * g(x) with respect to x?
f'(x) * g(x) + g'(x) * f(x)
If f and g are differentiable functions at x and g(x) ≠ 0, according to Theorem 3.3.6, what is the derivative of f(x)/g(x) with respect to x?
(g(x)*f'(x) - f(x)*g'(x))/g(x)^2
According to Theorem 3.3.7, what is the derivative of 1/g(x) with respect to x?
-g'(x)/g(x)^2
What is the derivative of sin(x) with respect to x?
cos(x)
What is the derivative of cos(x) with respect to x?
-sin(x)
What is the derivative of tan(x) with respect to x?
sec^2(x)
According to Theorem 3.3.8, what is the derivative of x^n with respect to x, where n is any integer?
nx^(n-1)
Which theorem generalizes the Power Rule (Theorem 3.3.1) for all integers (negative or non-negative)?
Theorem 3.3.8
What does Theorem 3.3.6 state about the differentiability of the quotient function f/g at x when g(x) ≠ 0?
'f/g' is differentiable at x and its derivative is (gf' - fg')/g^2.
What is the derivative of the constant function f(x) = 9?
0
If f(x) = $x^3$, what is the derivative f'(x)?
$3x^2$
What is the derivative of the function g(x) = $4x^2$?
$8x$
According to Theorem 3.3.6, what is the derivative of the function $\frac{1}{h(x)}$ with respect to x, where h(x) ≠ 0?
$-\frac{1}{h^2(x)}$
What is the derivative of the function $f(x) + g(x)$ with respect to x?
$f'(x) + g'(x)$
If f(x) = 7, what is the derivative f'(x)?
$0$
What is the derivative of the sum f(x) + g(x) according to Theorem 3.3.5?
f'(x) + g'(x)
When g(x) ≠ 0, what is the derivative of the quotient f(x)/g(x) according to Theorem 3.3.6?
(g(x)f'(x) - f(x)g'(x)) / [g(x)]^2
What is the derivative of sin(x) with respect to x?
cos(x)
According to Theorem 3.3.8, what is the derivative of x^n with respect to x, where n is any integer?
nx^(n-1)
What is the derivative of tan(x) with respect to x?
sec^2(x)
What is the derivative of the function g(x) = sin(2x)?
-2cos(2x)
According to Theorem 3.3.7, what is the derivative of 1/g(x) with respect to x when g(x) ≠ 0?
-[g'(x)] / [g(x)]^2
What is the derivative of cos(x) with respect to x?
-sin(x)
According to Theorem 3.3.5, what is the derivative of f(x) * g(x) with respect to x?
[f'(x)]g(x) + f(x)[g'(x)]
What does Theorem 3.3.6 state about the differentiability of the quotient function f/g at x when g(x) ≠ 0?
(f/g)' = (gf' - fg') / [g]^2
Explore the theorems and shortcuts for calculating derivatives of special functions including constant functions, power functions, and constant multiples of functions. Learn about the derivative of constant functions and how to apply Theorem 3.3.1.
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