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Math  /  Calculus

Questionf′(x)=4x−10f^{\prime}(x)=4 x-10
Finding the Derivative of a Simple Rational function Que if f(x)=1x(x≠0)f(x)=\frac{1}{x}(x \neq 0) find f′(x)f^{\prime}(x) Que. If f(x)=xf(x)=\sqrt{x}, ford f′(x)f^{\prime}(x) Now f(x)=1xf(x)=\frac{1}{x}

Studdy Solution

STEP 1

1. We are given the function f(x)=1x f(x) = \frac{1}{x} where x≠0 x \neq 0 .
2. We need to find the derivative f′(x) f^{\prime}(x) .

STEP 2

1. Rewrite the function in a form that is easier to differentiate.
2. Apply the power rule to find the derivative.
3. Simplify the derivative expression.

STEP 3

Rewrite the function f(x)=1x f(x) = \frac{1}{x} as a power of x x :
f(x)=x−1 f(x) = x^{-1}

STEP 4

Apply the power rule for differentiation, which states that if f(x)=xn f(x) = x^n , then f′(x)=nxn−1 f^{\prime}(x) = nx^{n-1} .
For f(x)=x−1 f(x) = x^{-1} :
f′(x)=(−1)x−1−1 f^{\prime}(x) = (-1)x^{-1-1} f′(x)=−x−2 f^{\prime}(x) = -x^{-2}

STEP 5

Rewrite the derivative in a more standard form:
f′(x)=−1x2 f^{\prime}(x) = -\frac{1}{x^2}
The derivative of f(x)=1x f(x) = \frac{1}{x} is:
f′(x)=−1x2 f^{\prime}(x) = -\frac{1}{x^2}

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