Power Rule with Polynomial Expressions

To find the derivative of f(x) = x⁷, we apply the power rule. The power rule states that if f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹. In this case, n = 7, so f'(x) = 7·x^(7-1) = 7x⁶.

To find the derivative of g(x) = 5x³, 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 x³ using the power rule: d/dx(x³) = 3x². Then we multiply by the constant 5: g'(x) = 5·3x² = 15x².

To find the derivative of h(x) = x^(-3), we apply the power rule. The power rule states that if h(x) = xⁿ, then h'(x) = n·xⁿ⁻¹. In this case, n = -3, so h'(x) = -3·x^(-3-1) = -3x^(-4).

To find the derivative of f(x) = 2x⁴ - 3x³ + 5x - 8, 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(2x⁴) = 2·4x³ = 8x³. For the second term: d/dx(-3x³) = -3·3x² = -9x². For the third term: d/dx(5x) = 5. For the constant term: d/dx(-8) = 0. Combining these, we get f'(x) = 8x³ - 9x² + 5.

First, we rewrite g(x) = 3/x² as g(x) = 3x^(-2). 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 = -2, so d/dx(x^(-2)) = -2x^(-3). Then we multiply by the constant 3: g'(x) = 3·(-2x^(-3)) = -6x^(-3). Rewriting in fraction form, we get g'(x) = -6/x³.

First, we rewrite h(x) = 4√x as h(x) = 4x^(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 4: h'(x) = 4·(½)x^(-½) = 2x^(-½). Rewriting in radical form, we get h'(x) = 2/√x.

To find the derivative of f(x) = 4x⁵ - 2x³ + 7x - 3, we apply the power rule to each term separately and use the sum/difference rule. For the first term: d/dx(4x⁵) = 4·5x⁴ = 20x⁴. For the second term: d/dx(-2x³) = -2·3x² = -6x². For the third term: d/dx(7x) = 7. For the constant term: d/dx(-3) = 0. Combining these, we get f'(x) = 20x⁴ - 6x² + 7.

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

To find the derivative of h(x) = 6x^(-5), 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 = -5, so d/dx(x^(-5)) = -5x^(-6). Then we multiply by the constant 6: h'(x) = 6·(-5x^(-6)) = -30x^(-6).

To find the derivative of f(x) = 3x⁶ - 4x⁴ + 2x³ - 7x + 2, we apply the power rule to each term separately and use the sum/difference rule. For the first term: d/dx(3x⁶) = 3·6x⁵ = 18x⁵. For the second term: d/dx(-4x⁴) = -4·4x³ = -16x³. For the third term: d/dx(2x³) = 2·3x² = 6x². For the fourth term: d/dx(-7x) = -7. For the constant term: d/dx(2) = 0. Combining these, we get f'(x) = 18x⁵ - 16x³ + 6x² - 7.

First, we find the derivative of f(x) = 3x⁴ 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 = 4, so d/dx(x⁴) = 4x³. Then we multiply by the constant 3: f'(x) = 3·4x³ = 12x³. Now we evaluate f'(2) = 12·(2)³ = 12·8 = 96.

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 = x³ using the power rule: dy/dx = 3x². Then we evaluate the derivative at x = 2: dy/dx|_(x=2) = 3·(2)² = 3·4 = 12. Therefore, the slope of the tangent line at the point (2, 8) is 12.

To find the derivative of g(x) = 2x⁵ + 4x³ - 3x² + 6x - 5, 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(4x³) = 4·3x² = 12x². For the third term: d/dx(-3x²) = -3·2x = -6x. For the fourth term: d/dx(6x) = 6. For the constant term: d/dx(-5) = 0. Combining these, we get g'(x) = 10x⁴ + 12x² - 6x + 6.

To find the derivative of h(x) = 5x^(⅓), 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 = 1/3, so d/dx(x^(⅓)) = (⅓)x^(-2/3). Then we multiply by the constant 5: h'(x) = 5·(⅓)x^(-2/3) = (5/3)x^(-2/3).

First, we find the derivative of f(x) = 3x⁴ - 2x³ + 4x² - 5x + 1 using the power rule and 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(4x²) = 4·2x = 8x. For the fourth term: d/dx(-5x) = -5. For the constant term: d/dx(1) = 0. Combining these, we get f'(x) = 12x³ - 6x² + 8x - 5. Now we evaluate f'(1) = 12·(1)³ - 6·(1)² + 8·1 - 5 = 12 - 6 + 8 - 5 = 9.