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The derivative of the function is dy/dx = 4([tex]x^{4sinx}[/tex])[(sin x) / x + (cos x) (ln x)]
y = [tex]x^{4sin x}[/tex]
Taking the log of both sides:
ln y = sin x ln [tex]x^{4}[/tex] = (sin x) * (4 ln x) = 4 (sin x)(ln x)
Now differentiate both sides. On the left you'll need to use the chain rule, and on the right you'll use the product rule:
1/y dy/dx = 4[(sin x) (1/x) + (cos x)(ln x)] = 4 [(sin x) / x + (cos x)(ln x)]
Multiply both sides by y
dy/dx = y * 4 [(sin x) / x + (cos x)(ln x)]
Since y = [tex]x^{4sinx}[/tex], we can rewrite this as:
dy/dx = [tex]x^{4sinx}[/tex] * 4 [(sin x) / x + (cos x)(ln x)]
dy/dx = 4[tex]x^{4sinx}[/tex] [(sin x) / x + (cos x)(ln x)]
Chain rule is the formula used to find the derivative of a composite function. Product rule is used to find derivative of products of two or more functions.
Therefore, the derivative of the function y = [tex]x^{4sin x}[/tex] is dy/dx = 4([tex]x^{4sinx}[/tex])[(sin x) / x + (cos x) (ln x)]
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