The process of differentiating a function often involves applying various rules of differentiation, such as the chain rule, power rule, and constant multiple rule. For example, when differentiating the function f(x) = ln(2x² – 3x + 4), we first apply the chain rule to differentiate the outside function while keeping the inside function unchanged. Then, we set up the derivative of the inside function, applying the constant multiple rule to each term and using the power rule to differentiate each term. Finally, we simplify the expressions to obtain the result (4x – 3) / (2x² – 3x + 4). This step-by-step process helps to break down complex functions into manageable parts, making it easier to understand and apply the principles of differentiation.
\( f(x) = \sin(\sqrt{x}) \) – Start with the given function.
\( \frac{d}{dx} \sin(\sqrt{x}) = \cos(\sqrt{x}) \cdot \frac{d}{dx} x^{1/2} \) – Apply the chain rule: differentiate the outer function.
\( = \cos(\sqrt{x}) \cdot \frac{1}{2} x^{1/2 – 1} \) – Differentiate the inner function using the power rule.
\( = \cos(\sqrt{x}) \cdot \frac{1}{2} x^{-1/2} \) – Simplify the exponent.
\( = \cos(\sqrt{x}) \cdot \frac{1}{2} \cdot \frac{1}{x^{1/2}} \) – Rewrite as a product of fractions.
\( = \cos(\sqrt{x}) \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{x}} \) – Express \( x^{-1/2} \) as \( \frac{1}{\sqrt{x}} \).
\( = \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \) – Combine the fractions.
\( = \frac{\cos(\sqrt{x})}{2\sqrt{x}} \) – Write \(\cos(\sqrt{x})\) over \(2\sqrt{x}\) to show the result.
\( f(x) = \cos(2x^2) \) – Start with the given function.
\( \frac{d}{dx} \cos(2x^2) = -\sin(2x^2) \cdot \frac{d}{dx} 2x^2 \) – Apply the chain rule: differentiate the outer function.
\( = -\sin(2x^2) \cdot 2 \cdot \frac{d}{dx} x^2 \) – Apply the constant multiple rule.
\( = -\sin(2x^2) \cdot 2 \cdot 2x \) – Differentiate the inner function using the power rule.
\( = -4x \sin(2x^2) \) – Multiply the expressions to obtain the result.
\( f(x) = \tan(x^3) \) – Start with the given function.
\( \frac{d}{dx} \tan(x^3) = \sec^2(x^3) \cdot \frac{d}{dx} x^3 \) – Differentiate the outside function, copy the inside unchanged, and set up the derivative of the inside.
\( = \sec^2(x^3) \cdot 3x^2 \) – Differentiate the inside function using the power rule.
\( = 3x^2 \sec^2(x^3) \) – Multiply the expressions to obtain the result.
\( f(x) = \ln(x^2 + 1) \) – Start with the given function.
\( \frac{d}{dx} \ln(x^2 + 1) = \frac{1}{x^2 + 1} \cdot \frac{d}{dx} (x^2 + 1) \) – Differentiate the outside function, copy the inside unchanged, and set up the derivative of the inside.
\( = \frac{1}{x^2 + 1} \cdot (2x) \) – Differentiate the inside function using the power rule.
\( = \frac{2x}{x^2 + 1} \) – Multiply the expressions to obtain the result.
\( f(x) = e^{3x^2 – 4x} \) – Start with the given function.
\( \frac{d}{dx} e^{3x^2 – 4x} = e^{3x^2 – 4x} \cdot \frac{d}{dx} (3x^2 – 4x) \) – Differentiate the outside function, copy the inside unchanged, and set up the derivative of the inside.
\( = e^{3x^2 – 4x} \cdot (6x – 4) \) – Differentiate the inside function using the power rule.
\( = (6x – 4) e^{3x^2 – 4x} \) – Multiply the expressions to obtain the result.
\( f(x) = \sqrt{4x^3 – 2x^2 + x} \) – Start with the given function.
\( \frac{d}{dx} \sqrt{4x^3 – 2x^2 + x} = \frac{1}{2\sqrt{4x^3 – 2x^2 + x}} \cdot \frac{d}{dx} (4x^3 – 2x^2 + x) \) – Apply the chain rule: differentiate the outside function and set up the derivative of the inside.
\( = \frac{1}{2\sqrt{4x^3 – 2x^2 + x}} \cdot \left( 4 \cdot \frac{d}{dx} x^3 – 2 \cdot \frac{d}{dx} x^2 + \frac{d}{dx} x \right) \) – Apply the constant multiple rule to each term inside the parentheses.
\( = \frac{1}{2\sqrt{4x^3 – 2x^2 + x}} \cdot \left( 4 \cdot 3x^2 – 2 \cdot 2x + 1 \right) \) – Differentiate each term inside the parentheses using the power rule.
\( = \frac{1}{2\sqrt{4x^3 – 2x^2 + x}} \cdot \left( 4 \cdot 3 \cdot x^2 – 2 \cdot 2 \cdot x + 1 \right) \) – Expand the multiplication inside the parentheses.
\( = \frac{1}{2\sqrt{4x^3 – 2x^2 + x}} \cdot \left( 12x^2 – 4x + 1 \right) \) – Simplify the expressions inside the parentheses.
\( = \frac{12x^2 – 4x + 1}{2\sqrt{4x^3 – 2x^2 + x}} \) – Combine the fractions to obtain the result.
\( f(x) = \sin(5x^2 – 3x + 2) \) – Start with the given function.
\( \frac{d}{dx} \sin(5x^2 – 3x + 2) = \cos(5x^2 – 3x + 2) \cdot \frac{d}{dx} (5x^2 – 3x + 2) \) – Apply the chain rule: differentiate the outside function, copy the inside unchanged, and set up the derivative of the inside.
\( = \cos(5x^2 – 3x + 2) \cdot \left( 5 \cdot \frac{d}{dx} x^2 – 3 \cdot \frac{d}{dx} x + \frac{d}{dx} 2 \right) \) – Apply the constant multiple rule to each term inside the parentheses.
\( = \cos(5x^2 – 3x + 2) \cdot \left( 5 \cdot 2x – 3 \cdot 1 + 0 \right) \) – Differentiate each term inside the parentheses using the power rule and the constant rule.
\( = \cos(5x^2 – 3x + 2) \cdot \left( 10x – 3 + 0 \right) \) – Simplify the expressions inside the parentheses.
\( = \cos(5x^2 – 3x + 2) \cdot (10x – 3) \) – Remove the unnecessary zero.
\( = (10x – 3) \cos(5x^2 – 3x + 2) \) – Multiply the expressions to obtain the result.
\( f(x) = \frac{1}{3x^3 – 2x + 7} \) – Start with the given function.
\( f(x) = (3x^3 – 2x + 7)^{-1} \) – Rewrite the function using a negative exponent.
\( \frac{d}{dx} (3x^3 – 2x + 7)^{-1} \) – Begin the differentiation process.
\( = -1 \cdot (3x^3 – 2x + 7)^{-2} \cdot \frac{d}{dx} (3x^3 – 2x + 7) \) – Apply the chain rule and power rule to the outside function, copy the inside unchanged, and set up the derivative of the inside.
\( = -\frac{1}{(3x^3 – 2x + 7)^2} \cdot \left( 3 \cdot 3x^2 – 2 \cdot 1 + 0 \right) \) – Differentiate the inside function using the power rule and constant rule, and rewrite the expression as a fraction.
\( = -\frac{9x^2 – 2}{(3x^3 – 2x + 7)^2} \) – Simplify the expressions to obtain the result.
\( f(x) = \ln(2x^2 – 3x + 4) \) – Start with the given function.
\( \frac{d}{dx} \ln(2x^2 – 3x + 4) \) – Begin the differentiation process.
\( = \frac{1}{2x^2 – 3x + 4} \cdot \frac{d}{dx} (2x^2 – 3x + 4) \) – Apply the chain rule: differentiate the outside function, copy the inside unchanged, and set up the derivative of the inside.
\( = \frac{1}{2x^2 – 3x + 4} \cdot \left( 2 \cdot \frac{d}{dx} x^2 – 3 \cdot \frac{d}{dx} x + \frac{d}{dx} 4 \right) \) – Apply the constant multiple rule to each term inside the parentheses.
\( = \frac{1}{2x^2 – 3x + 4} \cdot \left( 2 \cdot 2x – 3 \cdot 1 + 0 \right) \) – Differentiate each term inside the parentheses using the power rule and the constant rule.
\( = \frac{1}{2x^2 – 3x + 4} \cdot (4x – 3) \) – Simplify the expressions inside the parentheses.
\( = \frac{4x – 3}{2x^2 – 3x + 4} \) – Multiply the expressions to obtain the result.