Find the Derivative - d/dx y=(7-x^2)(x^3-x+5)

Problem

d()/d(x)*(7−x2)*(x3−x+5)

Solution

1. Identify the product rule for differentiation, which states that for two functions u and v d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the expression to u and v such that u=7−x2 and v=x3−x+5

3. Differentiate each part individually to find d(u)/d(x)=−2*x and d(v)/d(x)=3*x2−1

4. Apply the formula by substituting these components into the product rule expression: (7−x2)*(3*x2−1)+(x3−x+5)*(−2*x)

5. Expand the first term using the FOIL method: 21*x2−7−3*x4+x2

6. Expand the second term by distributing −2*x −2*x4+2*x2−10*x

7. Combine like terms to simplify the expression: (−3*x4−2*x4)+(21*x2+x2+2*x2)−10*x−7

8. Simplify the final polynomial to get −5*x4+24*x2−10*x−7

Final Answer

d()/d(x)*(7−x2)*(x3−x+5)=−5*x4+24*x2−10*x−7

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