Math  /  Calculus

QuestionEvaluate f(x)f^{\prime}(x) at x=1x=1.  Evaluate f(x) at x=1.f(1)=\begin{array}{l} \text { Evaluate } f^{\prime}(x) \text { at } x=1 . \\ f^{\prime}(1)=\square \end{array}

Studdy Solution

STEP 1

1. We are given the function f(x)=4x8x2+2x8 f(x) = \frac{4\sqrt{x}}{8x^2 + 2x - 8} .
2. We need to find the derivative f(x) f'(x) and evaluate it at x=1 x = 1 .

STEP 2

1. Differentiate the function f(x) f(x) using the quotient rule.
2. Simplify the expression for f(x) f'(x) .
3. Substitute x=1 x = 1 into the derivative.
4. Evaluate the expression to find f(1) f'(1) .

STEP 3

Differentiate f(x)=4x8x2+2x8 f(x) = \frac{4\sqrt{x}}{8x^2 + 2x - 8} using the quotient rule:
The quotient rule states that if f(x)=u(x)v(x) f(x) = \frac{u(x)}{v(x)} , then
f(x)=u(x)v(x)u(x)v(x)(v(x))2 f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
Here, u(x)=4x u(x) = 4\sqrt{x} and v(x)=8x2+2x8 v(x) = 8x^2 + 2x - 8 .

STEP 4

Find u(x) u'(x) :
u(x)=4x=4x1/2 u(x) = 4\sqrt{x} = 4x^{1/2} u(x)=412x1/2=2x1/2=2x u'(x) = 4 \cdot \frac{1}{2} x^{-1/2} = 2x^{-1/2} = \frac{2}{\sqrt{x}}

STEP 5

Find v(x) v'(x) :
v(x)=8x2+2x8 v(x) = 8x^2 + 2x - 8 v(x)=16x+2 v'(x) = 16x + 2

STEP 6

Substitute u(x) u'(x) , u(x) u(x) , v(x) v'(x) , and v(x) v(x) into the quotient rule formula:
f(x)=(2x)(8x2+2x8)(4x)(16x+2)(8x2+2x8)2 f'(x) = \frac{\left(\frac{2}{\sqrt{x}}\right)(8x^2 + 2x - 8) - (4\sqrt{x})(16x + 2)}{(8x^2 + 2x - 8)^2}

STEP 7

Simplify the expression for f(x) f'(x) :
First, simplify the numerator:
2x(8x2+2x8)=16x2+4x16x \frac{2}{\sqrt{x}}(8x^2 + 2x - 8) = \frac{16x^2 + 4x - 16}{\sqrt{x}}
4x(16x+2)=64x3/2+8x1/2 4\sqrt{x}(16x + 2) = 64x^{3/2} + 8x^{1/2}
Combine the terms:
f(x)=16x2+4x16(64x3/2+8x1/2)(8x2+2x8)2 f'(x) = \frac{16x^2 + 4x - 16 - (64x^{3/2} + 8x^{1/2})}{(8x^2 + 2x - 8)^2}

STEP 8

Substitute x=1 x = 1 into the simplified expression for f(x) f'(x) :
f(1)=16(1)2+4(1)16(64(1)3/2+8(1)1/2)(8(1)2+2(1)8)2 f'(1) = \frac{16(1)^2 + 4(1) - 16 - (64(1)^{3/2} + 8(1)^{1/2})}{(8(1)^2 + 2(1) - 8)^2}

STEP 9

Simplify the expression:
f(1)=16+416(64+8)(8+28)2 f'(1) = \frac{16 + 4 - 16 - (64 + 8)}{(8 + 2 - 8)^2}
f(1)=47222 f'(1) = \frac{4 - 72}{2^2}
f(1)=684 f'(1) = \frac{-68}{4}

STEP 10

Evaluate the expression:
f(1)=17 f'(1) = -17
The value of f(1) f'(1) is:
17 \boxed{-17}

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