Q. Given that w=y3+5, find dyd(5w4+4cosy) in terms of only y.Answer:
Given function and task: Given the function w=y3+5, we need to find the derivative of 5w4+4cos(y) with respect to y. We will use the chain rule for differentiation, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Find derivative of w: First, let's find the derivative of w with respect to y. Since w=y3+5, we have:dydw=dyd(y3+5)dydw=3y2+0dydw=3y2
Differentiate 5w4: Next, we need to differentiate the function 5w4 with respect to w and then multiply by the derivative of w with respect to y. This is an application of the chain rule.dwd(5w4)=20w3Then, we multiply by dydw:dyd(5w4)=20w3×dydwdyd(5w4)=20w3×3y2
Substitute w in terms of y: Now, we need to substitute w back in terms of y to express the derivative in terms of y only.w=y3+5dyd(5w4)=20(y3+5)3⋅3y2
Differentiate 4cos(y): We also need to differentiate the term 4cos(y) with respect to y.dyd(4cos(y))=−4sin(y)
Add derivatives: Finally, we add the derivatives of the two terms to find the derivative of the entire function with respect to y. dyd(5w4+4cos(y))=dyd(5w4)+dyd(4cos(y)) dyd(5w4+4cos(y))=20(y3+5)3⋅3y2−4sin(y)
Simplify final answer: Simplify the expression to get the final answer. dyd(5w4+4cos(y))=60y2(y3+5)3−4sin(y)
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