/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 Use any method to evaluate the d... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 71

Use any method to evaluate the derivative of the following functions. $$h(x)=\left(5 x^{7}+5 x\right)\left(6 x^{3}+3 x^{2}+3\right)$$

Short Answer

Expert verified
Based on the given function h(x) = (5x^7 + 5x)(6x^3 + 3x^2 + 3), we used the product rule to find its derivative. After finding the derivatives of the individual functions and applying the product rule, we simplified the expression to get the final derivative. The derivative of the function h(x), denoted as h'(x), is: h'(x) = 300x^9 + 135x^8 + 105x^6 + 120x^3 + 45x^2 + 15

Step by step solution

01

Finding f'(x)

To find the derivative of the function f(x) = 5x^7 + 5x, use the power rule for differentiation. The power rule states that if you have a function ax^n, then its derivative is given by anx^(n-1), where a and n are constants. So, for f(x): $$f'(x) = \frac{d}{dx}(5x^7) + \frac{d}{dx}(5x)$$ Applying the power rule: $$f'(x) = (5)(7)x^{7-1} + (5)(1)x^{1-1}$$ Simplify: $$f'(x) = 35x^6 + 5$$
02

Finding g'(x)

Next, find the derivative of the function g(x) = 6x^3 + 3x^2 + 3 using the power rule for differentiation. $$g'(x) = \frac{d}{dx}(6x^3) + \frac{d}{dx}(3x^2) + \frac{d}{dx}(3)$$ Applying the power rule: $$g'(x) = (6)(3)x^{3-1} + (3)(2)x^{2-1} + 0$$ Simplify: $$g'(x) = 18x^2 + 6x$$
03

Applying the product rule

Now that we have f'(x) and g'(x), we can apply the product rule to find h'(x): h'(x) = f'(x)g(x) + f(x)g'(x) Substitute f'(x), g(x), f(x) and g'(x) into the formula: $$h'(x) = (35x^6 + 5)(6x^3 + 3x^2 + 3) + (5x^7 + 5x)(18x^2 + 6x)$$
04

Simplifying h'(x)

We now need to simplify the expression h'(x). First, distribute the terms inside the parentheses: $$h'(x) = [35x^6(6x^3) + 35x^6(3x^2) + 35x^6(3) + 5(6x^3) + 5(3x^2) + 5(3)]$$ $$ + [5x^7(18x^2) + 5x^7(6x) + 5x(18x^2) + 5x(6x)]$$ Next, multiply the terms: $$h'(x) = [210x^9 + 105x^8 + 105x^6 + 30x^3 + 15x^2 + 15]$$ $$ + [90x^9 + 30x^8 + 90x^3 + 30x^2]$$ Finally, combine like terms: $$h'(x) = [210x^9 + 90x^9] + [105x^8 + 30x^8] + [105x^6] + [30x^3+90x^3] + [15x^2+30x^2] +[15]$$ $$h'(x) = 300x^9 + 135x^8 + 105x^6 + 120x^3 + 45x^2 +15$$ The derivative of the given function, h'(x), is: $$h'(x) = 300x^9 + 135x^8 + 105x^6 + 120x^3 + 45x^2 +15$$

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