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How to use the Chain Rule together with Product Rule or Quotient Rule?

Question:

How to use the Chain Rule together with Product Rule or Quotient Rule?

Methods of Differentiation:

{eq}\\ {/eq}

  • Chain Rule - To differentiate a composite function i.e. functions of the form {eq}F=f(g(x)) {/eq}, we use chain rule of differentiation as {eq}F'=f'(g(x)).g'(x) {/eq}
  • For two differentiable functions, the product rule can be applied as {eq}g'(x)h(x)+g(x)h'(x). {/eq}
  • The quotient rule for {eq}F=\left(\dfrac{g(x)}{h(x)}\right) {/eq} can be applied as {eq}F'=\dfrac{g'(x)h(x)-g(x)h'(x)}{g^2(x)} {/eq}

Answer and Explanation:

{eq}\\ {/eq}

  • Consider three differentiable functions {eq}f(x), g(x) \ \& \ h(x) {/eq}, related by {eq}F(x)=f(g(x).h(x)) {/eq}, we can apply chain rule and product rule as follows :-

{eq}F'(x)=f'(g(x).h(x)).(g'(x)h(x)+g(x)h'(x)) {/eq}

  • Now, again Consider three differentiable functions {eq}f(x), g(x) \ \& \ h(x) {/eq}, related by {eq}F(x)=f\left(\dfrac{g(x)}{h(x)}\right) {/eq}, we can apply chain rule and quotient rule as follows :-

{eq}F'(x)=f'(g(x).h(x)).\left(\dfrac{g'(x)h(x)-g(x)h'(x)}{g^2(x)}\right) {/eq}


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When to Use the Quotient Rule for Differentiation

from Math 104: Calculus

Chapter 8 / Lesson 8
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