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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac { 5 x } { 24 x - 46 x ^ { 2 } - 8 x + 66 x + 26 x + 42 }$$

Evaluate

$\frac{5x}{2\left(21+54x-23x^{2}\right)}$

Short Solution Steps

Combine $24x$ and $-8x$ to get $16x$.

$$\frac{5x}{16x-46x^{2}+66x+26x+42}$$

Combine $16x$ and $66x$ to get $82x$.

$$\frac{5x}{82x-46x^{2}+26x+42}$$

Combine $82x$ and $26x$ to get $108x$.

$$\frac{5x}{108x-46x^{2}+42}$$

Differentiate w.r.t. x

$-\frac{5\left(23x^{2}+21\right)}{2\left(-529x^{4}+2484x^{3}-1950x^{2}-2268x-441\right)}$

Steps Using Derivative Rule for Quotient

Combine $24x$ and $-8x$ to get $16x$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{16x-46x^{2}+66x+26x+42})$$

Combine $16x$ and $66x$ to get $82x$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{82x-46x^{2}+26x+42})$$

Combine $82x$ and $26x$ to get $108x$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{108x-46x^{2}+42})$$

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.

$$\frac{\left(108x^{1}-46x^{2}+42\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1})-5x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(108x^{1}-46x^{2}+42)}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

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 $nax^{n-1}$.

$$\frac{\left(108x^{1}-46x^{2}+42\right)\times 5x^{1-1}-5x^{1}\left(108x^{1-1}+2\left(-46\right)x^{2-1}\right)}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

Simplify.

$$\frac{\left(108x^{1}-46x^{2}+42\right)\times 5x^{0}-5x^{1}\left(108x^{0}-92x^{1}\right)}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

Multiply $108x^{1}-46x^{2}+42$ times $5x^{0}$.

$$\frac{108x^{1}\times 5x^{0}-46x^{2}\times 5x^{0}+42\times 5x^{0}-5x^{1}\left(108x^{0}-92x^{1}\right)}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

Multiply $5x^{1}$ times $108x^{0}-92x^{1}$.

$$\frac{108x^{1}\times 5x^{0}-46x^{2}\times 5x^{0}+42\times 5x^{0}-\left(5x^{1}\times 108x^{0}+5x^{1}\left(-92\right)x^{1}\right)}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

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

$$\frac{108\times 5x^{1}-46\times 5x^{2}+42\times 5x^{0}-\left(5\times 108x^{1}+5\left(-92\right)x^{1+1}\right)}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

Simplify.

$$\frac{540x^{1}-230x^{2}+210x^{0}-\left(540x^{1}-460x^{2}\right)}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

Combine like terms.

$$\frac{230x^{2}+210x^{0}}{\left(108x^{1}-46x^{2}+42\right)^{2}}$$

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

$$\frac{230x^{2}+210x^{0}}{\left(108x-46x^{2}+42\right)^{2}}$$

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

$$\frac{230x^{2}+210\times 1}{\left(108x-46x^{2}+42\right)^{2}}$$

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

$$\frac{230x^{2}+210}{\left(108x-46x^{2}+42\right)^{2}}$$