Practice Questions (with Step-by-Step Solutions) Question 1. Differentiate [y = x^{x}], [x>0]. Step-by-Step Solution: Take natural log of both sides:…
1. Why Logarithmic Differentiation? Some functions are difficult to differentiate directly, such as: Functions with variables in both base and
1. What is an Implicit Function? A function where y is not explicitly isolated in the form [y = f(x)],
Practice Questions (with Step-by-Step Solutions) Question 1. Differentiate [f(x)=x^{3} + 4x^{2} – 5x + 7]. Step-by-Step Solution: This is a
Practice Questions (with Step-by-Step Solutions) Question 1. Differentiate [f(x)=(5x+3)^{4}]. Step-by-Step Solution: Identify inner and outer: inner [g(x)=5x+3], outer [F(u)=u^{4}]. [\dfrac{d}{du}(u^{4})=4u^{3}].
1. Concept Overview If a function is a composition of two (or more) functions, say [y = f(g(x))], then the
Question 1. Differentiate [f(x)][=\dfrac{x^{2}+1}{x+1}] , [x\ne -1]. Step-by-Step Solution: Let [u=x^{2}+1], [v=x+1]. Compute derivatives: [u’=2x], [v’=1]. Apply quotient rule: [\dfrac{v
1. Concept Overview When a function is written as a quotient of two differentiable functions: [f(x)=\dfrac{u(x)}{v(x)}] Quotient Rule Formula [f'(x)][=\dfrac{v(x)\cdot
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