Solve f(x)=x/x+1quad[32]

$$f ( x ) = \frac { x } { x + 1 } \quad [ 32 ]$$

Differentiate w.r.t. x

$$\frac{32}{\left(x+1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express \frac{x}{x+1}\times 32 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\times 32}{x+1})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(x^{1}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(32x^{1})-32x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}+1)}{\left(x^{1}+1\right)^{2}}$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{\left(x^{1}+1\right)\times 32x^{1-1}-32x^{1}x^{1-1}}{\left(x^{1}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(x^{1}+1\right)\times 32x^{0}-32x^{1}x^{0}}{\left(x^{1}+1\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{1}\times 32x^{0}+32x^{0}-32x^{1}x^{0}}{\left(x^{1}+1\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{32x^{1}+32x^{0}-32x^{1}}{\left(x^{1}+1\right)^{2}}$$

Combine like terms.

$$\frac{\left(32-32\right)x^{1}+32x^{0}}{\left(x^{1}+1\right)^{2}}$$

Subtract 32 from 32.

$$\frac{32x^{0}}{\left(x^{1}+1\right)^{2}}$$

For any term t, t^{1}=t.

$$\frac{32x^{0}}{\left(x+1\right)^{2}}$$

For any term t except 0, t^{0}=1.

$$\frac{32\times 1}{\left(x+1\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{32}{\left(x+1\right)^{2}}$$

Evaluate

$$\frac{32x}{x+1}$$

Solution Steps

Express \frac{x}{x+1}\times 32 as a single fraction.

$$\frac{x\times 32}{x+1}$$