Successive Differentiation Problems with Solutions

 step-by-step solution for successive differentiation problems in a clear and simple manner.

Problem 1: Successive Differentiation of a Polynomial

Problem: Find the first, second, and third derivatives of the function $f(x) = 3x^4 - 5x^3 + 2x - 7$.

Solution:

1. First Derivative:

   • Write the function: $f(x) = 3x^4 - 5x^3 + 2x - 7$
   • Differentiate each term:
     • The derivative of $3x^4$ is $12x^3$
     • The derivative of $-5x^3$ is $-15x^2$
     • The derivative of $2x$ is $2$
     • The derivative of $-7$ is $0$ (since the derivative of a constant is zero)
   • Combine the results: $$f'(x) = 12x^3 - 15x^2 + 2$$

2. Second Derivative:

   • Write the first derivative: $f'(x) = 12x^3 - 15x^2 + 2$
   • Differentiate each term:
     • The derivative of $12x^3$ is $36x^2$
     • The derivative of $-15x^2$ is $-30x$
     • The derivative of $2$ is $0$
   • Combine the results: $$f''(x) = 36x^2 - 30x$$

3. Third Derivative:

   • Write the second derivative: $f''(x) = 36x^2 - 30x$
   • Differentiate each term:
     • The derivative of $36x^2$ is $72x$
     • The derivative of $-30x$ is $-30$
   • Combine the results: $$f'''(x) = 72x - 30$$

Problem 2: Successive Differentiation of an Exponential Function

Problem: Find the first and second derivatives of the function $g(x) = e^{2x}$.

Solution:

1. First Derivative:

   • Write the function: $g(x) = e^{2x}$
   • Use the chain rule. The outer function is $e^u$ where $u = 2x$:
     • The derivative of $e^u$ is $e^u$
     • Multiply by the derivative of $u$: $\frac{d}{dx}(2x) = 2$
   • Combine the results: $$g'(x) = e^{2x} \cdot 2 = 2e^{2x}$$

2. Second Derivative:

   • Write the first derivative: $g'(x) = 2e^{2x}$
   • Use the chain rule again. The outer function is $e^u$ where $u = 2x$:
     • The derivative of $e^u$ is $e^u$
     • Multiply by the derivative of $u$: $\frac{d}{dx}(2x) = 2$
   • Combine the results: $$g''(x) = 2 \cdot e^{2x} \cdot 2 = 4e^{2x}$$

Problem 3: Successive Differentiation of a Trigonometric Function

Problem: Find the first, second, and third derivatives of the function $h(x) = \sin(x)$.

Solution:

1. First Derivative:

   • Write the function: $h(x) = \sin(x)$
   • The derivative of $\sin(x)$ is $\cos(x)$
   • Combine the results: $$h'(x) = \cos(x)$$

2. Second Derivative:

   • Write the first derivative: $h'(x) = \cos(x)$
   • The derivative of $\cos(x)$ is $-\sin(x)$
   • Combine the results: $$h''(x) = -\sin(x)$$

3. Third Derivative:

   • Write the second derivative: $h''(x) = -\sin(x)$
   • The derivative of $-\sin(x)$ is $-\cos(x)$
   • Combine the results: $$h'''(x) = -\cos(x)$$

Problem 4: Successive Differentiation of a Logarithmic Function

Problem: Find the first and second derivatives of the function $k(x) = \ln(x)$.

Solution:

1. First Derivative:

   • Write the function: $k(x) = \ln(x)$
   • The derivative of $\ln(x)$ is $\frac{1}{x}$
   • Combine the results: $$k'(x) = \frac{1}{x}$$

2. Second Derivative:

• Write the first derivative: $k'(x) = \frac{1}{x}$
• Rewrite $\frac{1}{x}$ as $x^{-1}$
• The derivative of $x^{-1}$ is $-x^{-2}$
• Combine the results: $$k''(x) = -\frac{1}{x^2}$$​

Problem 5: Successive Differentiation of a Polynomial

Problem: Find the first, second, and third derivatives of the function $f(x) = 4x^5 - 3x^4 + x^2 - 6$.

Solution:

1. First Derivative:

   • Write the function: $f(x) = 4x^5 - 3x^4 + x^2 - 6$
   • Differentiate each term:
     • The derivative of $4x^5$ is $20x^4$
     • The derivative of $-3x^4$ is $-12x^3$
     • The derivative of $x^2$ is $2x$
     • The derivative of $-6$ is $0$ (since the derivative of a constant is zero)
   • Combine the results: $$f'(x) = 20x^4 - 12x^3 + 2x$$

2. Second Derivative:

   • Write the first derivative: $f'(x) = 20x^4 - 12x^3 + 2x$
   • Differentiate each term:
     • The derivative of $20x^4$ is $80x^3$
     • The derivative of $-12x^3$ is $-36x^2$
     • The derivative of $2x$ is $2$
   • Combine the results: $$f''(x) = 80x^3 - 36x^2 + 2$$

3. Third Derivative:

   • Write the second derivative: $f''(x) = 80x^3 - 36x^2 + 2$
   • Differentiate each term:
     • The derivative of $80x^3$ is $240x^2$
     • The derivative of $-36x^2$ is $-72x$
     • The derivative of $2$ is $0$
   • Combine the results: $$f'''(x) = 240x^2 - 72x$$

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Problem 6: Successive Differentiation of an Exponential Function

Problem: Find the first and second derivatives of the function $g(x) = e^{3x}$.

Solution:

1. First Derivative:

   • Write the function: $g(x) = e^{3x}$
   • Use the chain rule. The outer function is $e^u$ where $u = 3x$:
     • The derivative of $e^u$ is $e^u$
     • Multiply by the derivative of $u$: $\frac{d}{dx}(3x) = 3$
   • Combine the results: $$g'(x) = e^{3x} \cdot 3 = 3e^{3x}$$

2. Second Derivative:

   • Write the first derivative: $g'(x) = 3e^{3x}$
   • Use the chain rule again. The outer function is $e^u$ where $u = 3x$:
     • The derivative of $e^u$ is $e^u$
     • Multiply by the derivative of $u$: $\frac{d}{dx}(3x) = 3$
   • Combine the results: $$g''(x) = 3 \cdot e^{3x} \cdot 3 = 9e^{3x}$$

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Problem 7: Successive Differentiation of a Trigonometric Function

Problem: Find the first, second, and third derivatives of the function $h(x) = \cos(x)$.

Solution:

1. First Derivative:

   • Write the function: $h(x) = \cos(x)$
   • The derivative of $\cos(x)$ is $-\sin(x)$
   • Combine the results: $$h'(x) = -\sin(x)$$

2. Second Derivative:

   • Write the first derivative: $h'(x) = -\sin(x)$
   • The derivative of $-\sin(x)$ is $-\cos(x)$
   • Combine the results: $$h''(x) = -\cos(x)$$

3. Third Derivative:

   • Write the second derivative: $h''(x) = -\cos(x)$
   • The derivative of $-\cos(x)$ is $\sin(x)$
   • Combine the results: $$h'''(x) = \sin(x)$$

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Problem 8: Successive Differentiation of a Logarithmic Function

Problem: Find the first and second derivatives of the function $k(x) = \ln(3x)$.

Solution:

1. First Derivative:

   • Write the function: $k(x) = \ln(3x)$
   • Use the chain rule. The outer function is $\ln(u)$ where $u = 3x$:
     • The derivative of $\ln(u)$ is $\frac{1}{u}$
     • Multiply by the derivative of $u$: $\frac{d}{dx}(3x) = 3$
   • Combine the results: $$k'(x) = \frac{1}{3x} \cdot 3 = \frac{3}{3x} = \frac{1}{x}$$

2. Second Derivative:

   • Write the first derivative: $k'(x) = \frac{1}{x}$
   • Rewrite $\frac{1}{x}$ as $x^{-1}$
   • The derivative of $x^{-1}$ is $-x^{-2}$
   • Combine the results: $$k''(x) = -\frac{1}{x^2}$$

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Problem 9: Successive Differentiation of a Rational Function

Problem: Find the first and second derivatives of the function $m(x) = \frac{1}{x+1}$.

Solution:

1. First Derivative:

   • Write the function: $m(x) = \frac{1}{x+1}$
   • Rewrite $\frac{1}{x+1}$ as $(x+1)^{-1}$
   • Use the power rule and chain rule:
     • The derivative of $(x+1)^{-1}$ is $-1 \cdot (x+1)^{-2}$
     • Multiply by the derivative of $x+1$, which is 1
   • Combine the results: $$m'(x) = -\frac{1}{(x+1)^2}$$

2. Second Derivative:

   • Write the first derivative: $m'(x) = -\frac{1}{(x+1)^2}$
   • Rewrite $-\frac{1}{(x+1)^2}$ as $-(x+1)^{-2}$
   • Use the power rule and chain rule again:
     • The derivative of $-(x+1)^{-2}$ is $2(x+1)^{-3}$
     • Multiply by the derivative of $x+1$, which is 1
   • Combine the results: $$m''(x) = \frac{2}{(x+1)^3}$$

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Problem 10: Successive Differentiation of a Combined Function

Problem: Find the first and second derivatives of the function $n(x) = x^2 \cdot \ln(x)$.

Solution:

1. First Derivative:

   • Write the function: $n(x) = x^2 \cdot \ln(x)$
   • Use the product rule: $(uv)' = u'v + uv'$
     • Let $u = x^2$ and $v = \ln(x)$
     • The derivative of $u = x^2$ is $u' = 2x$
     • The derivative of $v = \ln(x)$ is $v' = \frac{1}{x}$
   • Combine the results: $$n'(x) = (2x) \cdot \ln(x) + (x^2) \cdot \frac{1}{x} = 2x \ln(x) + x$$

2. Second Derivative:

   • Write the first derivative: $n'(x) = 2x \ln(x) + x$
   • Differentiate each term using the product rule again for $2x \ln(x)$:
     • Let $u = 2x$ and $v = \ln(x)$
     • The derivative of $u = 2x$ is $u' = 2$
     • The derivative of $v = \ln(x)$ is $v' = \frac{1}{x}$
     • Combine the results for the first term: $$(2x \ln(x))' = (2) \cdot \ln(x) + (2x) \cdot \frac{1}{x} = 2 \ln(x) + 2$$
     • The derivative of the second term $x$ is $1$
   • Combine all results: $$n''(x) = 2 \ln(x) + 2 + 1 = 2 \ln(x) + 3$$

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These problems cover a variety of functions, providing a thorough practice for students to understand the process of successive differentiation. If you have any more specific requests or need additional problems, feel free to ask!