VU 16-Techniques of Differentiation Lecture #16
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Questions and Answers

What is the derivative of the constant function f(x) = 7?

  • Undefined
  • 1
  • 7
  • 0 (correct)

    • According to Theorem 3.3.2, what is the derivative of $x^4$?

  • $3x^4$
  • $5x^3$
  • $4x^4$
  • $4x^3$ (correct)

    • If f(x) = 3, what is the derivative f'(x)?

  • 3
  • 1
  • 3x
  • 0 (correct)

    • What is the derivative of the function g(x) = 2x?

    <p>2x</p> Signup and view all the answers

    According to Theorem 3.3.1, what is the derivative of the constant function f(x) = -9?

    <p>0</p> Signup and view all the answers

    What is the derivative of the function h(x) = 5x^2?

    <p>$10x$</p> Signup and view all the answers

    What is the derivative of f(x) + g(x) with respect to x?

    <p>f'(x) + g'(x)</p> Signup and view all the answers

    According to Theorem 3.3.5, what is the derivative of f(x) * g(x) with respect to x?

    <p>f'(x) * g(x) + g'(x) * f(x)</p> Signup and view all the answers

    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?

    <p>(g(x)*f'(x) - f(x)*g'(x))/g(x)^2</p> Signup and view all the answers

    According to Theorem 3.3.7, what is the derivative of 1/g(x) with respect to x?

    <p>-g'(x)/g(x)^2</p> Signup and view all the answers

    What is the derivative of sin(x) with respect to x?

    <p>cos(x)</p> Signup and view all the answers

    What is the derivative of cos(x) with respect to x?

    <p>-sin(x)</p> Signup and view all the answers

    What is the derivative of tan(x) with respect to x?

    <p>sec^2(x)</p> Signup and view all the answers

    According to Theorem 3.3.8, what is the derivative of x^n with respect to x, where n is any integer?

    <p>nx^(n-1)</p> Signup and view all the answers

    Which theorem generalizes the Power Rule (Theorem 3.3.1) for all integers (negative or non-negative)?

    <p>Theorem 3.3.8</p> Signup and view all the answers

    What does Theorem 3.3.6 state about the differentiability of the quotient function f/g at x when g(x) ≠ 0?

    <p>'f/g' is differentiable at x and its derivative is (gf' - fg')/g^2.</p> Signup and view all the answers

    What is the derivative of the constant function f(x) = 9?

    <p>0</p> Signup and view all the answers

    If f(x) = $x^3$, what is the derivative f'(x)?

    <p>$3x^2$</p> Signup and view all the answers

    What is the derivative of the function g(x) = $4x^2$?

    <p>$8x$</p> Signup and view all the answers

    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?

    <p>$-\frac{1}{h^2(x)}$</p> Signup and view all the answers

    What is the derivative of the function $f(x) + g(x)$ with respect to x?

    <p>$f'(x) + g'(x)$</p> Signup and view all the answers

    If f(x) = 7, what is the derivative f'(x)?

    <p>$0$</p> Signup and view all the answers

    What is the derivative of the sum f(x) + g(x) according to Theorem 3.3.5?

    <p>f'(x) + g'(x)</p> Signup and view all the answers

    When g(x) ≠ 0, what is the derivative of the quotient f(x)/g(x) according to Theorem 3.3.6?

    <p>(g(x)f'(x) - f(x)g'(x)) / [g(x)]^2</p> Signup and view all the answers

    What is the derivative of sin(x) with respect to x?

    <p>cos(x)</p> Signup and view all the answers

    According to Theorem 3.3.8, what is the derivative of x^n with respect to x, where n is any integer?

    <p>nx^(n-1)</p> Signup and view all the answers

    What is the derivative of tan(x) with respect to x?

    <p>sec^2(x)</p> Signup and view all the answers

    What is the derivative of the function g(x) = sin(2x)?

    <p>-2cos(2x)</p> Signup and view all the answers

    According to Theorem 3.3.7, what is the derivative of 1/g(x) with respect to x when g(x) ≠ 0?

    <p>-[g'(x)] / [g(x)]^2</p> Signup and view all the answers

    What is the derivative of cos(x) with respect to x?

    <p>-sin(x)</p> Signup and view all the answers

    According to Theorem 3.3.5, what is the derivative of f(x) * g(x) with respect to x?

    <p>[f'(x)]<em>g(x) + f(x)</em>[g'(x)]</p> Signup and view all the answers

    What does Theorem 3.3.6 state about the differentiability of the quotient function f/g at x when g(x) ≠ 0?

    <p>(f/g)' = (g<em>f' - f</em>g') / [g]^2</p> Signup and view all the answers

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