Differentiate the following functions with respect to x. a) 2√x b) ln(3x + 2) c) x² + y² + 2x + y = 0 d) cos 6x e) 3 sin 2x

To differentiate 2√(x) with respect to x: Step 1: Rewrite the function using exponent notation. y = 2√(x) = 2x^(1/2); Step 2: Apply the power rule for differentiation, which states that d/dx(axⁿ) = anxⁿ⁻¹. dy/dx = 2 × 1/2 x^(1/2 − 1) Step 3: Simplify the expression. dy/dx = 1 × x^(−1/2); dy/dx = x^(−1/2) Step 4: Express the result with a positive exponent and in radical form. dy/dx = 1x^(1/2) = (1)/(√(x)) The derivative of 2√(x) with respect to x is (1)/(√(x)).

Differentiate the following functions with respect to x. a) 2√x b) ln(3x + 2) c) x² + y² + 2x + y = 0 d) cos 6x e) 3 sin 2x

To differentiate 2√(x) with respect to x: Step 1: Rewrite the function using exponent notation. y = 2√(x) = 2x^(1/2); Step 2: Apply the power rule for differentiation, which states that d/dx(axⁿ) = anxⁿ⁻¹. dy/dx = 2 × 1/2 x^(1/2 − 1) Step 3: Simplify the expression. dy/dx = 1 × x^(−1/2); dy/dx = x^(−1/2) Step 4: Express the result with a positive exponent and in radical form. dy/dx = 1x^(1/2) = (1)/(√(x)) The derivative of 2√(x) with respect to x is (1)/(√(x)).

Solved: Differentiate the following functions with respect…

To differentiate $2\sqrt{x}$ with respect to $x$ :

Rewrite the function using exponent notation.
$y = 2\sqrt{x} = 2x^{\frac{1}{2}}$
Apply the power rule for differentiation, which states that $\frac{d}{dx}(ax^n) = anx^{n-1}$ .
$\frac{dy}{dx} = 2 \times \frac{1}{2} x^{\frac{1}{2} - 1}$
Simplify the expression.
$\frac{dy}{dx} = 1 \times x^{-\frac{1}{2}}$ $\frac{dy}{dx} = x^{-\frac{1}{2}}$
Express the result with a positive exponent and in radical form.
$\frac{dy}{dx} = \frac{1}{x^{\frac{1}{2}}} = \frac{1}{\sqrt{x}}$
The derivative of $2\sqrt{x}$ with respect to $x$ is $\boxed{\frac{1}{\sqrt{x}}}$ .

Differentiate the following functions with respect to x. a) 2√x b) ln(3x + 2) c) x² + y² + 2x + y = 0 d) cos 6x e) 3 sin 2x

Step-by-step solution

Rewrite the function using exponent notation.

Apply the power rule for differentiation, which states that $\frac{d}{dx}(ax^n) = anx^{n-1}$ .

Simplify the expression.

Express the result with a positive exponent and in radical form.

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More Mathematics Questions

Differentiate the following functions with respect to x. a) 2√x b) ln(3x + 2) c) x² + y² + 2x + y = 0 d) cos 6x e) 3 sin 2x

Step-by-step solution

Rewrite the function using exponent notation.

Apply the power rule for differentiation, which states that ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}dxd​(axn)=anxn−1.

Simplify the expression.

Express the result with a positive exponent and in radical form.

Still stuck on this one?

More Mathematics Questions

Apply the power rule for differentiation, which states that $\frac{d}{dx}(ax^n) = anx^{n-1}$ .