Differentiation Using the Chain Rule SOLUTION 1 : Differentiate. $$\displaystyle y=\cos \left( {4x} \right)$$, $$\displaystyle g\left( x \right)=\cos \left( {\tan x} \right)$$, $$\displaystyle \begin{array}{l}f\left( x \right)={{\sec }^{3}}\left( {\pi x} \right)\\f\left( x \right)={{\left[ {\sec \left( {\pi x} \right)} \right]}^{3}}\end{array}$$, $$\displaystyle \begin{array}{l}f\left( \theta \right)=2{{\cot }^{2}}\left( {2\theta } \right)+\theta \\f\left( \theta \right)=2{{\left[ {\cot \left( {2\theta } \right)} \right]}^{2}}+\theta \end{array}$$. Evaluate any superscripted expression down to a single number before evaluating the power. In the next section, we use the Chain Rule to justify another differentiation technique. Example 1 Find the derivative f '(x), if f is given by f(x) = 4 cos (5x - 2) Solution to Example 1 Let u = 5x - 2 and f(u) = 4 cos u, hence du / dx = 5 and df / du = - 4 sin u We now use the chain rule eval(ez_write_tag([[580,400],'shelovesmath_com-medrectangle-4','ezslot_2',110,'0','0']));Understand these problems, and practice, practice, practice! An expression in an exponent (a small, raised number indicating a power) groups that expression like parentheses do. Yes, sometimes we have to use the chain rule twice, in the cases where we have a function inside a function inside another function. Anytime there is a parentheses followed by an exponent is the general rule of thumb. Use the Product Rule, since we have $$t$$âs in both expressions. \displaystyle \begin{align}{l}{g}â\left( x \right)&=\frac{1}{4}{{\left( {\color{red}{{16-{{x}^{3}}}}} \right)}^{{-\frac{3}{4}}}}\cdot \left( {\color{red}{{-3{{x}^{2}}}}} \right)\\&=-\frac{{3{{x}^{2}}}}{{4{{{\left( {16-{{x}^{3}}} \right)}}^{{\frac{3}{4}}}}}}=-\frac{{3{{x}^{2}}}}{{4\,\sqrt[4]{{{{{\left( {16-{{x}^{3}}} \right)}}^{3}}}}}}\end{align}. When should you use the Chain Rule? 4. The reason we also took out a $$\frac{3}{2}$$ is because itâs the GCF of $$\frac{3}{2}$$ and $$\frac{{24}}{2}\,\,(12)$$. \displaystyle \begin{align}{f}â\left( x \right)&=3\,{{\color{red}{{\sec }}}^{2}}\left( {\color{blue}{{\pi x}}} \right)\cdot \left( {\color{red}{{\sec \left( {\color{blue}{{\pi x}}} \right)\tan \left( {\color{blue}{{\pi x}}} \right)}}} \right)\color{blue}{\pi }\\&=3\pi {{\sec }^{3}}\left( {\pi x} \right)\tan \left( {\pi x} \right)\end{align}, This oneâs a little tricky, since we have to use the Chain Rule, \displaystyle \begin{align}{f}â\left( \theta \right)=&4\,\color{red}{{\cot }}\left( {\color{blue}{{2\theta }}} \right)\cdot \color{red}{{-{{{\csc }}^{2}}\left( {\color{blue}{{2\theta }}} \right)}}\cdot \color{blue}{2}+1\\&=1-8{{\csc }^{2}}\left( {2\theta } \right)\cot \left( {2\theta } \right)\end{align}. Since the last step is multiplication, we treat the express Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another and then both of them are inside a third function (you can have four or more nested functions, but three is probably the most you’ll see). This is more formally stated as, if the functions f (x) and g (x) are both differentiable and define F (x) = (f o g)(x), then the required derivative of the function F(x) is, This formal approach … ... To evaluate the expression above you (1) evaluate the expression inside the parentheses and the (2) raise that result to the 53 power. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. Browse other questions tagged derivatives chain-rule transcendental-equations or ask your own question. $$\displaystyle \begin{array}{l}{y}â=-\sin \left( {\color{red}{{4x}}} \right)\cdot \color{red}{4}\\=-4\sin \left( {4x} \right)\end{array}$$, Since the $$\left( {4x} \right)$$ is the inner function (the argument of $$\text{sin}$$), we have to take multiply by the derivative of that function, which is, \displaystyle \begin{align}{g}â\left( x \right)&=-\sin \left( {\color{red}{{\tan x}}} \right)\cdot \color{red}{{{{{\sec }}^{2}}x}}\\&=-{{\sec }^{2}}x\cdot \sin \left( {\tan x} \right)\end{align}. The equation of the tangent line to $$f\left( \theta \right)=\cos \left( {5\theta } \right)$$ at the point $$\displaystyle \left( {\frac{\pi }{2},0} \right)$$ is $$\displaystyle y=-5x+\frac{{5\pi }}{2}$$. are the inner functions, we have to multiply each by their derivative. The chain rule says when weâre taking the derivative, if thereâs something other than $$\boldsymbol {x}$$ (like in parentheses or under a radical sign) when weâre using one of the rules weâve learned (like the power rule), we have to multiply by the derivative of whatâs in the parentheses. $$\begin{array}{c}f\left( x \right)={{\left( {5{{x}^{4}}-2} \right)}^{3}}\\x=1\end{array}$$, $$\displaystyle {f}â\left( x \right)=3{{\left( {5{{x}^{4}}-2} \right)}^{2}}\left( {20{{x}^{3}}} \right)=60{{x}^{3}}{{\left( {5{{x}^{4}}-2} \right)}^{2}}$$. As an example, let's analyze 4•(x³+5)² Speaking informally we could say the "inside function" is (x 3 +5) and the "outside function" is 4 • (inside) 2. Take a look at the same example listed above. 312. f (x) = (2 x3 + 1) (x5 – x) The chain rule is a rule, in which the composition of functions is differentiable. $${p}â\left( 4 \right)\text{ and }{q}â\left( {-1} \right)$$, The Equation of the Tangent Line with the Chain Rule, \displaystyle \begin{align}{f}â\left( x \right)&=8{{\left( {\color{red}{{5x-1}}} \right)}^{7}}\cdot \color{red}{5}\\&=40{{\left( {5x-1} \right)}^{7}}\end{align}, Since the $$\left( {5x-1} \right)$$ is the inner function, after using the Power Rule, we have to multiply by the derivative of that function, which is, \displaystyle \begin{align}{f}â\left( x \right)&=3{{\left( {\color{red}{{{{x}^{4}}-1}}} \right)}^{2}}\cdot \left( {\color{red}{{4{{x}^{3}}}}} \right)\\&=12{{x}^{3}}{{\left( {{{x}^{4}}-1} \right)}^{2}}\end{align}. So let’s dive right into it! At point $$\left( {1,27} \right)$$, the slope is $$\displaystyle 60{{\left( 1 \right)}^{3}}{{\left[ {5{{{\left( 1 \right)}}^{4}}-2} \right]}^{2}}=540$$. 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