Find the value of df?¹/dx at x = f(a). 9A 1 f(x) = ?x + 6, a = 3 A) 1/6 9B f(x) = 4x², x ? 0, a = 2 A) 1/16 9C f(x) = x³ – 9x² – 1, x ? 6, a = 5 A) -15 Find the formula for df?¹/dx. 9D 1 5 f(x) = ?x + ? 8 16 A) 8x – 5/2 9E f(x) = (8 – x)³ A) x²/³ 9F f(x) = x?/³ A) x²/³ Solve the problem. 9G Find the derivative of the inverse of the function f(x) = mx, where m is a nonzero constant. A) mx²/2 B) 5 C) 6 B) 16 C) 3/32 B) -1 C) -1/101 B) x – 5/2 C) 8 B) 8 – x¹/³ C) -1/(3x²/³) B) x³/? C) 3x?²/? B) m C) 1/m D) 1/5 D) 1/8 D) -1/15 D) 1/8 D) -3(8 – x)² D) x²/? D) 1
The Correct Answer and Explanation is:
Let’s break this down step by step for each question and clarify how to approach them.
9A:
Function: f(x)=x+6f(x) = \sqrt{x + 6}f(x)=x+6, where a=3a = 3a=3.
We are tasked with finding ddx\frac{d}{dx}dxd of the function at x=f(a)x = f(a)x=f(a), meaning at x=3x = 3x=3.
- Find the derivative of f(x)f(x)f(x): f′(x)=ddx(x+6)=12x+6.f'(x) = \frac{d}{dx} \left( \sqrt{x + 6} \right) = \frac{1}{2\sqrt{x + 6}}.f′(x)=dxd(x+6)=2x+61.
- Evaluate at x=3x = 3x=3: f′(3)=123+6=129=16.f'(3) = \frac{1}{2\sqrt{3 + 6}} = \frac{1}{2\sqrt{9}} = \frac{1}{6}.f′(3)=23+61=291=61. So, the answer is A) 16\frac{1}{6}61.
9B:
Function: f(x)=4x2f(x) = 4x^2f(x)=4×2, where a=2a = 2a=2.
We are tasked with finding f′(x)f'(x)f′(x) at x=2x = 2x=2.
- Find the derivative of f(x)f(x)f(x): f′(x)=ddx(4×2)=8x.f'(x) = \frac{d}{dx} \left( 4x^2 \right) = 8x.f′(x)=dxd(4×2)=8x.
- Evaluate at x=2x = 2x=2: f′(2)=8×2=16.f'(2) = 8 \times 2 = 16.f′(2)=8×2=16. So, the correct answer is B) 1/161/161/16.
9C:
Function: f(x)=x3−9×2−1f(x) = x^3 – 9x^2 – 1f(x)=x3−9×2−1, where a=5a = 5a=5.
We need to find f′(x)f'(x)f′(x) at x=5x = 5x=5.
- Find the derivative of f(x)f(x)f(x): f′(x)=ddx(x3−9×2−1)=3×2−18x.f'(x) = \frac{d}{dx} \left( x^3 – 9x^2 – 1 \right) = 3x^2 – 18x.f′(x)=dxd(x3−9×2−1)=3×2−18x.
- Evaluate at x=5x = 5x=5: f′(5)=3(5)2−18(5)=3(25)−90=75−90=−15.f'(5) = 3(5)^2 – 18(5) = 3(25) – 90 = 75 – 90 = -15.f′(5)=3(5)2−18(5)=3(25)−90=75−90=−15. So, the correct answer is C) -15.
9D:
Formula: You are asked to find the derivative of the inverse of a function.
Let’s consider the function f(x)=mxf(x) = mxf(x)=mx, where mmm is a nonzero constant.
- Find the inverse function:
Since f(x)=mxf(x) = mxf(x)=mx, the inverse of f(x)f(x)f(x) is f−1(y)=ymf^{-1}(y) = \frac{y}{m}f−1(y)=my. - Use the inverse function derivative rule:
The derivative of the inverse function is given by: (f−1(x))′=1f′(f−1(x)).\left( f^{-1}(x) \right)’ = \frac{1}{f'(f^{-1}(x))}.(f−1(x))′=f′(f−1(x))1. - Find f′(x)f'(x)f′(x): f′(x)=m.f'(x) = m.f′(x)=m.
- Substitute into the formula: (f−1(x))′=1m.\left( f^{-1}(x) \right)’ = \frac{1}{m}.(f−1(x))′=m1. So, the correct answer is C) 1m\frac{1}{m}m1.
Final Answer Summary:
- 9A: A) 16\frac{1}{6}61
- 9B: B) 1/161/161/16
- 9C: C) -15
- 9D: C) 1m\frac{1}{m}m1
This set of questions requires using basic differentiation rules and the formula for derivatives of inverse functions. It’s important to be familiar with the power rule, chain rule, and the concept of function inverses when solving such problems
