Find the first derivative dy/dx of y = 8sqrt(x).

Question

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Step 1: Find the first derivative dy/dx of y = 8√(x). First, rewrite y using exponent notation: y = 8x^(1/2). Now, differentiate using the power rule d/dx(xⁿ) = nxⁿ⁻¹: dy/dx = 8 · 1/2 x^(1/2−1) = 4x^(−1/2) Step 2: Find the second derivative (d²y)/(dx²). Differentiate dy/dx = 4x^(−1/2) again: (d²y)/(dx²) = 4 · (−1/2) x^(−1/2−1) = −2x^(−3/2); Step 3: Substitute y, dy/dx, and (d²y)/(dx²) into the left-hand side of the equation 4x² (d²y)/(dx²) + 4x dy/dx = y. The left-hand side (LHS) is: 4x² (−2x^(−3/2)) + 4x (4x^(−1/2)) Step 4: Simplify the expression. Multiply the terms: −8x^(2 − 3/2) + 16x^(1 − 1/2) Combine the exponents: −8x^(4/2 − 3/2) + 16x^(2/2 − 1/2) −8x^(1/2) + 16x^(1/2) Combine the coefficients: (−8+16)x^(1/2) = 8x^(1/2) Rewrite in radical form: 8√(x) Step 5: Compare the simplified LHS with y. We found that the LHS simplifies to 8√(x). Given that y = 8√(x). Therefore, 4x² (d²y)/(dx²) + 4x dy/dx = y is shown.

Find the first derivative (dy)/(dx) of y = 8sqrt(x).

Step-by-step solution

Find the first derivative $\frac{dy}{dx}$ of $y = 8\sqrt{x}$ .

Find the second derivative $\frac{d^2y}{dx^2}$ .

Substitute $y$ , $\frac{dy}{dx}$ , and $\frac{d^2y}{dx^2}$ into the left-hand side of the equation $4x^2 \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} = y$ .

Simplify the expression.

Compare the simplified LHS with $y$ .

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More Mathematics Questions

Find the first derivative (dy)/(dx) of y = 8sqrt(x).

Step-by-step solution

Find the first derivative dydx\frac{dy}{dx}dxdy​ of y=8xy = 8\sqrt{x}y=8x​.

Find the second derivative d2ydx2\frac{d^2y}{dx^2}dx2d2y​.

Substitute yyy, dydx\frac{dy}{dx}dxdy​, and d2ydx2\frac{d^2y}{dx^2}dx2d2y​ into the left-hand side of the equation 4x2d2ydx2+4xdydx=y4x^2 \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} = y4x2dx2d2y​+4xdxdy​=y.