Power Rule Basics

To find the derivative of f(x) = x⁵, we apply the power rule. The power rule states that if f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹. In this case, n = 5, so f'(x) = 5·x^(5-1) = 5x⁴.

To find the derivative of g(x) = 3x⁴, we apply the power rule combined with the constant multiple rule. The constant multiple rule states that if g(x) = c·f(x), then g'(x) = c·f'(x). First, we find the derivative of x⁴ using the power rule: d/dx(x⁴) = 4x³. Then we multiply by the constant 3: g'(x) = 3·4x³ = 12x³.

To find the derivative of h(x) = x^(-2), we apply the power rule. The power rule states that if h(x) = xⁿ, then h'(x) = n·xⁿ⁻¹. In this case, n = -2, so h'(x) = -2·x^(-2-1) = -2x^(-3).

To find the derivative of f(x) = 4x³ + 2x² - 5x + 7, we apply the power rule to each term separately and use the sum/difference rule. The sum/difference rule states that the derivative of a sum/difference is the sum/difference of the derivatives. For the first term: d/dx(4x³) = 4·3x² = 12x². For the second term: d/dx(2x²) = 2·2x = 4x. For the third term: d/dx(-5x) = -5·1 = -5. For the constant term: d/dx(7) = 0. Combining these, we get f'(x) = 12x² + 4x - 5.

First, we rewrite g(x) = 2/x³ as g(x) = 2x^(-3). Now we apply the power rule combined with the constant multiple rule. The power rule states that if g(x) = xⁿ, then g'(x) = n·xⁿ⁻¹. In this case, n = -3, so d/dx(x^(-3)) = -3x^(-4). Then we multiply by the constant 2: g'(x) = 2·(-3x^(-4)) = -6x^(-4). Rewriting in fraction form, we get g'(x) = -6/x⁴.

First, we rewrite h(x) = 3√x as h(x) = 3x^(½). Now we apply the power rule combined with the constant multiple rule. The power rule states that if h(x) = xⁿ, then h'(x) = n·xⁿ⁻¹. In this case, n = 1/2, so d/dx(x^(½)) = (½)x^(-½). Then we multiply by the constant 3: h'(x) = 3·(½)x^(-½) = (3/2)x^(-½). Rewriting in radical form, we get h'(x) = 3/(2√x).

To find the derivative of f(x) = 5x⁴ - 3x² + 6x - 2, we apply the power rule to each term separately and use the sum/difference rule. For the first term: d/dx(5x⁴) = 5·4x³ = 20x³. For the second term: d/dx(-3x²) = -3·2x = -6x. For the third term: d/dx(6x) = 6·1 = 6. For the constant term: d/dx(-2) = 0. Combining these, we get f'(x) = 20x³ - 6x + 6.

To find the derivative of g(x) = x³ + 4x² - 5, we apply the power rule to each term separately and use the sum/difference rule. For the first term: d/dx(x³) = 3x². For the second term: d/dx(4x²) = 4·2x = 8x. For the constant term: d/dx(-5) = 0. Combining these, we get g'(x) = 3x² + 8x.

To find the derivative of h(x) = 7x^(-4), we apply the power rule combined with the constant multiple rule. The power rule states that if h(x) = xⁿ, then h'(x) = n·xⁿ⁻¹. In this case, n = -4, so d/dx(x^(-4)) = -4x^(-5). Then we multiply by the constant 7: h'(x) = 7·(-4x^(-5)) = -28x^(-5).

To find the derivative of f(x) = 2x⁵ - 3x³ + x² - 4x + 1, we apply the power rule to each term separately and use the sum/difference rule. For the first term: d/dx(2x⁵) = 2·5x⁴ = 10x⁴. For the second term: d/dx(-3x³) = -3·3x² = -9x². For the third term: d/dx(x²) = 2x. For the fourth term: d/dx(-4x) = -4. For the constant term: d/dx(1) = 0. Combining these, we get f'(x) = 10x⁴ - 9x² + 2x - 4.

First, we find the derivative of f(x) = 4x³ using the power rule combined with the constant multiple rule. The power rule states that if f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹. In this case, n = 3, so d/dx(x³) = 3x². Then we multiply by the constant 4: f'(x) = 4·3x² = 12x². Now we evaluate f'(2) = 12·(2)² = 12·4 = 48.

To find the derivative of g(x) = 3x⁴ + 2x³ - 5x² + x - 7, we apply the power rule to each term separately and use the sum/difference rule. For the first term: d/dx(3x⁴) = 3·4x³ = 12x³. For the second term: d/dx(2x³) = 2·3x² = 6x². For the third term: d/dx(-5x²) = -5·2x = -10x. For the fourth term: d/dx(x) = 1. For the constant term: d/dx(-7) = 0. Combining these, we get g'(x) = 12x³ + 6x² - 10x + 1.

To find the derivative of h(x) = 4x^(½), we apply the power rule combined with the constant multiple rule. The power rule states that if h(x) = xⁿ, then h'(x) = n·xⁿ⁻¹. In this case, n = 1/2, so d/dx(x^(½)) = (½)x^(-½). Then we multiply by the constant 4: h'(x) = 4·(½)x^(-½) = 2x^(-½).

First, we find the derivative of f(x) = 2x³ - 3x² + 5x - 1 using the power rule and sum/difference rule. For the first term: d/dx(2x³) = 2·3x² = 6x². For the second term: d/dx(-3x²) = -3·2x = -6x. For the third term: d/dx(5x) = 5. For the constant term: d/dx(-1) = 0. Combining these, we get f'(x) = 6x² - 6x + 5. Now we evaluate f'(1) = 6·(1)² - 6·1 + 5 = 6 - 6 + 5 = 5.

The slope of the tangent line to a curve at a point is given by the derivative of the function at that point. First, we find the derivative of y = x² using the power rule: dy/dx = 2x. Then we evaluate the derivative at x = 3: dy/dx|_(x=3) = 2·3 = 6. Therefore, the slope of the tangent line at the point (3, 9) is 6.